Algebra and Fundamentals - Revision Notes
Algebra and Fundamentals is one of the highest-yield topics in the MSTE (Mathematics, Surveying, and Transportation Engineering) component of the PRC Civil Engineer Licensure Examination. Expect 8–12 questions per board exam covering quadratic and polynomial equations, arithmetic and geometric progressions, the binomial theorem, logarithms, and classic word problems (work, mixture, age, motion). This revision note condenses every exam-critical formula, decision rule, and worked example you need to maximize your score. Study the flowcharts carefully — they will help you select the correct formula under time pressure.
Laws of Exponents and Radicals
Formulas
a^m · a^n = a^(m+n)
Example
Simplify 3^4 · 3^(−2) = 3^(4+(−2)) = 3^2 = 9
Variables
a = base (real number, a ≠ 0); m, n = exponents (integers or rationals)
Application
Combining exponential expressions with the same base
a^(m/n) = (n-th root of a)^m
Example
8^(2/3) = (cube-root of 8)^2 = 2^2 = 4
Variables
a = radicand; m = power; n = index of the radical
Application
Converting between radical and exponential form
n-th root of (a^n) = |a| for even n; = a for odd n
Example
4th root of (x^4) = |x|; cube-root of (x^3) = x
Variables
a = any real number; n = index
Application
Simplifying even-indexed radicals to avoid sign errors
Exam tips
- When simplifying large exponents on the board exam, break them into factors of known values: 2^10 = 1024, 2^8 = 256.
- Convert all radical expressions to fractional exponents before applying algebra rules — it prevents index-handling errors.
- If the answer choices are numerical, substitute a simple value (e.g., a = 2) to verify your simplification.
- Expect 1–2 items per board exam on exponent/radical simplification. They are straightforward scoring opportunities — do not rush.
Key points
- Product rule: a^m · a^n = a^(m+n). The base must be the same.
- Power rule: (a^m)^n = a^(mn). Multiply the exponents.
- Quotient rule: a^m / a^n = a^(m−n).
- Zero exponent: a^0 = 1 for any a ≠ 0.
- Negative exponent: a^(−n) = 1/a^n. Move the base to the denominator.
- Fractional exponent: a^(m/n) = n-th root of a^m = (n-th root of a)^m.
- Radical product: n-th root of (ab) = (n-th root of a)(n-th root of b).
- Radical quotient: n-th root of (a/b) = (n-th root of a)/(n-th root of b).
- Rationalizing denominators — multiply numerator and denominator by the conjugate when a radical appears in the denominator.
- Simplify before solving: factor out perfect powers from under the radical sign to reduce arithmetic errors.
Definitions
Radicand
The expression under a radical sign.
Importance
Identifying the radicand correctly prevents incorrect simplification steps.
Rationalization
The process of eliminating radicals from the denominator of a fraction by multiplying by an appropriate conjugate or radical expression.
Importance
Board exam answer choices are always in rationalized form; failure to rationalize leads to mismatched answers.
Common mistakes
- Writing (a+b)^n = a^n + b^n — this is WRONG. The binomial theorem must be applied.
- Applying the product rule when bases differ: 2^3 · 3^2 ≠ 6^5.
- Forgetting the absolute value when simplifying even-indexed radicals: square-root of x^2 = |x|, not simply x.
- Negative exponent means reciprocal, NOT negative: 2^(−3) = 1/8, not −8.
- Confusing the index of a radical with the exponent of the radicand.
Logarithms
Formulas
log(MN) = log M + log N
Example
log(100 · 1000) = log 100 + log 1000 = 2 + 3 = 5. Verify: log(100,000) = log(10^5) = 5 ✓
Variables
M, N > 0; base is any positive number ≠ 1
Application
Expanding or condensing logarithmic expressions
log_b(x) = log(x) / log(b)
Example
log_2(32) = log(32)/log(2) = 1.50515/0.30103 = 5. Verify: 2^5 = 32 ✓
Variables
x > 0, b > 0, b ≠ 1
Application
Evaluating logarithms with non-standard bases on a scientific calculator
log M^p = p · log M
Example
Solve 3^x = 50: x·log 3 = log 50 → x = log(50)/log(3) = 1.69897/0.47712 ≈ 3.561
Variables
M > 0; p = any real number
Application
Solving exponential equations by taking logarithms of both sides
Exam tips
- Memorize: log 2 ≈ 0.3010, log 3 ≈ 0.4771, log 7 ≈ 0.8451. These values appear constantly in board exam computation.
- When the equation has logs on both sides with the same base, cancel the log: log_b(A) = log_b(B) → A = B.
- To solve 10^x = N, take log of both sides: x = log N. To solve e^x = N, take ln: x = ln N.
- In board exams, the word 'logarithm' without specifying a base always means base 10.
Key points
- Definition: log_b(x) = y means b^y = x. The logarithm is an exponent.
- Common log (log) uses base 10; natural log (ln) uses base e ≈ 2.71828.
- log(MN) = log M + log N — product rule.
- log(M/N) = log M − log N — quotient rule.
- log(M^p) = p·log M — power rule.
- Change of base: log_b(x) = ln(x)/ln(b) = log(x)/log(b).
- log_b(b) = 1; log_b(1) = 0; log_b(b^x) = x; b^(log_b x) = x.
- Logarithm is defined only for positive arguments: log(x) requires x > 0.
- When solving logarithmic equations, always check for extraneous solutions by substituting back into the original equation.
Definitions
Common Logarithm
Logarithm with base 10, written as log(x). It is the default base on most scientific calculators.
Importance
Used extensively in sound level (dB), pH, Richter scale, and board exam computation items.
Natural Logarithm
Logarithm with base e ≈ 2.71828, written as ln(x). It is the inverse of the exponential function e^x.
Importance
Appears in calculus-based engineering problems and growth/decay problems in board exams.
Antilogarithm
The inverse operation of a logarithm: if log(x) = y, then x = 10^y (antilog of y).
Importance
Required when solving for the argument after isolating the log side of an equation.
Common mistakes
- Writing log(M+N) = log M + log N — this is WRONG. The product rule applies to log(MN), not log(M+N).
- Writing log(M−N) = log M − log N — this is WRONG. The quotient rule applies to log(M/N).
- Dividing logs incorrectly: log(M)/log(N) ≠ log(M/N).
- Forgetting to check domain: log(x−3) requires x > 3, not x > 0.
- Applying change of base incorrectly: log_b(x) = log(x)/log(b), NOT log(b)/log(x).
Quadratic and Polynomial Equations
Formulas
x = [−b ± √(b² − 4ac)] / (2a)
Example
Solve 2x²+3x−5=0: a=2, b=3, c=−5. x=[−3±√(9+40)]/4=[−3±7]/4. x=(4/4)=1 or x=(−10/4)=−2.5
Variables
a = coefficient of x²; b = coefficient of x; c = constant term
Application
Finding roots of any quadratic equation; fallback when factoring is not immediately obvious
Sum = −b/a; Product = c/a
Example
For 2x²+3x−5=0: Sum=−3/2=−1.5 (check: 1+(−2.5)=−1.5 ✓); Product=−5/2=−2.5 (check: 1×(−2.5)=−2.5 ✓)
Variables
a, b, c from standard form ax²+bx+c=0
Application
Verifying computed roots; constructing a quadratic from known root properties
Remainder = f(k) when dividing f(x) by (x−k)
Example
Find the remainder when x³−3x²+2x−5 is divided by (x−2): f(2)=8−12+4−5=−5. Remainder=−5.
Variables
f(x) = polynomial; k = the value substituted
Application
Finding remainders without performing full polynomial division
Exam tips
- Try factoring first — it is faster. If b²−4ac is a perfect square, the quadratic factors over the integers.
- The sum/product of roots check takes 5 seconds and catches arithmetic errors before you bubble your answer.
- Board exam items on polynomials often ask for the remainder or the value of an unknown coefficient — use the Remainder or Factor Theorem directly.
- If asked for a quadratic equation with roots r₁ and r₂, write: x²−(r₁+r₂)x+(r₁·r₂)=0.
- Memorize: for equal roots (tangent to x-axis), set D=0. For real roots (crosses x-axis), set D≥0.
Key points
- Standard form: ax^2 + bx + c = 0, where a ≠ 0.
- Quadratic formula: x = [−b ± √(b²−4ac)] / (2a). This always works — memorize it.
- Discriminant D = b²−4ac determines the nature of roots.
- D > 0: two real, distinct roots. D = 0: one repeated real root. D < 0: two complex conjugate roots.
- Sum of roots: x₁ + x₂ = −b/a. Product of roots: x₁ · x₂ = c/a. These allow quick verification.
- Factoring: ax^2 + bx + c = a(x − x₁)(x − x₂) once roots are known.
- For a polynomial of degree n, there are exactly n roots (counting multiplicity) in the complex number system (Fundamental Theorem of Algebra).
- Remainder Theorem: when f(x) is divided by (x−k), the remainder = f(k).
- Factor Theorem: (x−k) is a factor of f(x) if and only if f(k) = 0.
- Synthetic division is the fastest way to evaluate polynomials and perform polynomial long division on board exams.
Definitions
Discriminant
The expression D = b²−4ac under the radical in the quadratic formula. Its sign determines the nature and number of real roots.
Importance
Board exam items frequently ask 'for what values of k does the equation have equal roots?' — set D=0 and solve for k.
Repeated Root
A root that occurs more than once, corresponding to a tangent point of the parabola to the x-axis. Occurs when D=0.
Importance
When D=0, the single root x=−b/(2a) is the vertex x-coordinate of the parabola.
Synthetic Division
A shortcut algorithm for dividing a polynomial by a linear factor (x−k), using only the coefficients.
Importance
Essential for the board exam — faster than long division and enables application of the Remainder and Factor Theorems.
Common mistakes
- Sign error in the quadratic formula: the formula starts with −b, not +b.
- Forgetting to check if the quadratic is in standard form (= 0) before applying the formula.
- Assuming integer roots when the discriminant is not a perfect square — leave answers as exact surds or decimals.
- In the Remainder Theorem, dividing by (x+k) means substituting x=−k, NOT x=+k.
- Dropping a coefficient of zero in synthetic division — always include placeholder zeros for missing degree terms.
Arithmetic and Geometric Progressions
Formulas
a_n = a₁ + (n−1)d
Example
AP: 3, 7, 11, 15,... Find a₁₀: a₁₀ = 3 + (10−1)(4) = 3 + 36 = 39
Variables
a_n = nth term; a₁ = first term; d = common difference; n = term number
Application
Finding any specific term in an arithmetic sequence
S_n = (n/2)(a₁ + a_n) = (n/2)[2a₁ + (n−1)d]
Example
Sum of first 10 terms, a₁=3, d=4: S₁₀=(10/2)[2(3)+9(4)]=5[6+36]=5(42)=210
Variables
S_n = sum of first n terms; a₁ = first term; a_n = last term; d = common difference
Application
Summing consecutive terms of an AP, including sum of first n positive integers
S_n = a₁(r^n − 1)/(r − 1)
Example
GP: 2, 6, 18,... Sum of first 5 terms: S₅ = 2(3⁵−1)/(3−1) = 2(243−1)/2 = 242
Variables
a₁ = first term; r = common ratio; n = number of terms
Application
Summing a finite geometric series
S∞ = a₁/(1−r), valid only when |r| < 1
Example
GP: 8, 4, 2, 1,... r=1/2. S∞ = 8/(1−1/2) = 8/(1/2) = 16
Variables
a₁ = first term; r = common ratio with |r| < 1
Application
Summing an infinite converging geometric series (e.g., repeating decimals, diminishing returns)
Exam tips
- Board exam problems often give you two non-consecutive AP terms and ask for the sum — set up two equations using a_n = a₁+(n−1)d and solve simultaneously.
- The sum of the first n positive integers: S = n(n+1)/2 — this is an AP with a₁=1 and d=1. Memorize this result.
- For infinite GP, always verify |r|<1 before applying S∞=a₁/(1−r). Board exams sometimes set a trap with r>1.
- Repeating decimals can be expressed as infinite GPs: 0.333... = 3/10 + 3/100 + ... = (3/10)/(1−1/10) = 1/3.
- Inserting k arithmetic means between a and b: the common difference d=(b−a)/(k+1).
Key points
- Arithmetic Progression (AP): each term is obtained by adding a constant d (common difference).
- AP general term: a_n = a₁ + (n−1)d.
- AP sum of n terms: S_n = (n/2)[2a₁ + (n−1)d] = (n/2)(a₁ + a_n).
- Arithmetic mean of two numbers a and b: AM = (a+b)/2.
- Geometric Progression (GP): each term is obtained by multiplying by a constant r (common ratio).
- GP general term: a_n = a₁ · r^(n−1).
- GP sum of n terms: S_n = a₁(r^n − 1)/(r − 1) for r ≠ 1.
- Infinite GP sum (converges only when |r| < 1): S∞ = a₁/(1−r).
- Geometric mean of two numbers a and b: GM = √(a·b) (positive root for positive terms).
- Harmonic Progression (HP): a sequence whose reciprocals form an AP. Solve HP problems by working with the AP of reciprocals.
- Relationship: AM ≥ GM ≥ HM (Arithmetic Mean − Geometric Mean − Harmonic Mean inequality).
Definitions
Common Difference (d)
The constant value added to each term of an AP to get the next term. d = a_n − a_(n−1).
Importance
Always compute d first in an AP problem. It unlocks both the general term and the sum formula.
Common Ratio (r)
The constant factor multiplied to each term of a GP to get the next term. r = a_n / a_(n−1).
Importance
Determining convergence of an infinite GP requires knowing |r|. If |r| ≥ 1, the infinite sum does not exist.
Harmonic Progression
A sequence of numbers whose reciprocals form an arithmetic progression.
Importance
Board exams often disguise HP problems by asking for the harmonic mean or the nth term — always convert to AP first.
Common mistakes
- Using the wrong 'n': ensure n is the number of TERMS, not the last term value.
- Applying the infinite GP formula when |r| ≥ 1 — the series diverges and S∞ does not exist.
- For GP problems, computing r by subtraction (like an AP) instead of division.
- In the AP sum formula, forgetting to multiply by n/2 or applying n instead of n/2.
- Confusing arithmetic mean (average) with geometric mean (square root of product) in mean-insertion problems.
Binomial Theorem
Formulas
T_(r+1) = C(n,r) · a^(n−r) · b^r
Example
Find the 6th term of (2x−1)^9: r=5, T₆=C(9,5)·(2x)^4·(−1)^5=126·16x⁴·(−1)=−2016x⁴
Variables
T_(r+1) = (r+1)th term; n = exponent; r = term index starting at 0; a, b = bases of the binomial
Application
Finding a specific term in a binomial expansion without expanding the entire expression
C(n,r) = n! / [r!(n−r)!]
Example
C(9,5) = 9!/(5!·4!) = (9·8·7·6)/(4·3·2·1) = 3024/24 = 126
Variables
n = total items; r = items chosen; ! = factorial
Application
Computing binomial coefficients; also used in probability and combinations
Exam tips
- To find the term containing x^k in (a+bx)^n, set the power of x in T_(r+1) equal to k and solve for r, then substitute back.
- For the term independent of x (constant term), set the power of x equal to zero and solve for r.
- In multiple-choice board exams, expand only the specific term asked — never expand the full binomial.
- Pascal's Triangle is fast for n ≤ 6; use the C(n,r) formula for n > 6.
- The sum of all binomial coefficients = 2^n. This is a quick sanity check: sum all your computed coefficients and verify.
Key points
- Binomial expansion: (a+b)^n = Σ C(n,k) · a^(n−k) · b^k for k=0 to n.
- C(n,k) = n! / [k!(n−k)!] is the binomial coefficient.
- The (r+1)th term (general term): T_(r+1) = C(n,r) · a^(n−r) · b^r.
- Total number of terms in the expansion of (a+b)^n is n+1.
- The middle term: for even n, there is one middle term at position (n/2)+1; for odd n, there are two middle terms.
- Pascal's Triangle gives the binomial coefficients for small values of n.
- The sum of all coefficients is obtained by substituting a=1, b=1: (1+1)^n = 2^n.
- The sum of coefficients of odd-positioned terms equals the sum of even-positioned terms = 2^(n−1).
Definitions
Binomial Coefficient C(n,r)
The number of ways to choose r objects from n objects, also equal to the coefficient of the (r+1)th term in the binomial expansion.
Importance
Board exam items on binomial theorem almost always require computing C(n,r) — practice evaluating it rapidly.
General Term of Binomial Expansion
The formula T_(r+1)=C(n,r)·a^(n−r)·b^r which gives the value of any specific term in (a+b)^n without full expansion.
Importance
This is the single most tested formula in binomial theorem board exam items. Know it cold.
Common mistakes
- Indexing error: the (r+1)th term has b raised to r (not r+1). T₆ has r=5 because T_(5+1)=T₆.
- Forgetting to apply the negative sign when b is negative: (2x−1)^9 means b=−1, not b=1.
- Computing C(n,r) with the wrong denominator — always use r! times (n−r)!.
- Confusing the term number with the value of r: the 4th term means r=3.
- Applying (a+b)^2 = a²+2ab+b² to higher powers — the middle terms will be wrong without using the theorem.
Word Problems — Work, Mixture, Age, and Motion
Formulas
Rate_combined = 1/t_A + 1/t_B − 1/t_C
Example
Pipe A fills in 3 h, Pipe B in 5 h, Drain C empties in 4 h. Rate = 1/3+1/5−1/4 = (20+12−15)/60 = 17/60. Time = 60/17 ≈ 3.53 h
Variables
t_A, t_B = time for A and B to finish alone (working); t_C = time for C to empty alone (draining); Rate_combined = net work rate
Application
Finding the combined rate when multiple workers/pipes work simultaneously, with one opposing
C₁V₁ + C₂V₂ = C_f(V₁+V₂)
Example
Mix 20 L of 30% acid with x L of 50% acid to get 40%. 0.30(20)+0.50(x)=0.40(20+x). 6+0.5x=8+0.4x. 0.1x=2. x=20 L
Variables
C₁, C₂ = concentrations of solutions being mixed; V₁, V₂ = volumes; C_f = final concentration
Application
Mixture problems involving two solutions combined to produce a desired concentration
D = R × T (Distance = Rate × Time)
Example
Two trains start from stations 300 km apart moving toward each other at 60 km/h and 90 km/h. Time to meet: 300/(60+90) = 300/150 = 2 h
Variables
D = distance (km or m); R = speed (km/h or m/s); T = time (h or s)
Application
All motion problems — rearrange as R=D/T or T=D/R as needed
Exam tips
- Work problems: always write the fraction of job done = Rate × Time. If the job is 'complete', set the fraction = 1.
- Board exam work problems with drain pipes: Rate_net = Rate_fill − Rate_drain. A negative result means the tank never fills.
- Motion problems: draw a simple sketch showing directions. Label it with D, R, T for each object.
- Age problems: build a table with columns 'Now' and 'x years ago/from now' — fill in all ages systematically.
- Mixture problems: always verify that V_final = V₁ + V₂ and C_final is between C₁ and C₂. If not, you set up the equation wrong.
Key points
- WORK PROBLEMS: Rate = 1/(time to complete alone). Combined rate = sum of individual rates. Time together = 1/(combined rate).
- For work problems, identify who is working WITH (rates add) and who is working AGAINST (rates subtract), e.g., a drain pipe subtracts.
- MIXTURE PROBLEMS: Amount of component in mixture = (concentration)(volume). Set up: initial + added = final.
- MOTION PROBLEMS: Distance = Rate × Time (D=RT). For two objects, identify whether they move toward each other (rates add), away from each other (rates add), or in the same direction (rates subtract).
- AGE PROBLEMS: Define the present age as the variable. Past age = present − years past. Future age = present + years future. Set up equations based on the stated relationships.
- COIN/NUMBER PROBLEMS: The value of coins = (number of coins)(face value). Sum all values to get total.
- Always define your variable clearly at the start of the solution. This is half the battle in word problems.
- After solving, substitute back into the original problem statement (not just the equation) to verify the answer makes physical sense.
Definitions
Work Rate
The fraction of a job completed per unit time, equal to 1/(time to complete alone). Unit: jobs per hour or jobs per day.
Importance
The fundamental concept of work problems. Add rates for workers doing the same job; subtract rates for opposing workers.
Concentration
The fraction (or percentage) of a pure substance in a mixture. Always expressed as a decimal in calculations (30% → 0.30).
Importance
Must be applied to the volume to get the amount of pure substance. Setting up the amount equation is the key step.
Common mistakes
- Adding times instead of rates in work problems: 'A takes 6 days and B takes 4 days, so together they take 10 days' is WRONG. Add rates: 1/6+1/4=5/12, so time=12/5=2.4 days.
- In mixture problems, writing the equation using percent instead of decimal (using 30 instead of 0.30).
- In motion problems, adding speeds when objects move in the same direction (should subtract) or subtracting when they move toward each other (should add).
- In age problems, not adjusting ALL ages consistently (past or future). If you go back 5 years, every person is 5 years younger.
- Setting up the equation without defining the unknown — leads to confusion and sign errors halfway through.
Connections
- Quadratic equations appear in structural engineering (beam deflection, projectile trajectory of falling debris) and in the NSCP load combination expressions.
- Geometric progressions model compound interest — directly applicable to engineering economics (PRC board exam topic) and present-worth/future-worth calculations.
- Logarithms are the mathematical foundation of the decibel scale (noise in construction sites), the Richter scale, and Manning's roughness formula logarithmic expressions.
- The Binomial Theorem is the basis of probability distributions (binomial distribution) used in quality control and reliability engineering.
- Work-rate problems are directly analogous to parallel resistance in electrical engineering and parallel flow paths in hydraulics (Pipes in parallel — add conductances, not resistances).
- Arithmetic progressions generalize to series solutions of differential equations encountered in Structural Analysis and Fluid Mechanics.
- The discriminant concept extends to the characteristic equation of a second-order ODE — critical damping (D=0), overdamping (D>0), underdamping (D<0).
- Sum and product of roots relate to Vieta's formulas, used in analysis of control systems and filter design in Electrical Engineering.
- Mixture problems are the algebraic model behind concrete mix design (water-cement ratio, aggregate proportioning) and soil stabilization problems.
- The change-of-base formula for logarithms is applied in the Hazen-Williams and Darcy-Weisbach equations when solving for friction factor iteratively.
Quick review questions
Simplify: 8^(2/3) × 4^(−1/2).
Answer
4 × (1/2) = 2
8^(2/3) = (cube-root of 8)^2 = 2^2 = 4. 4^(−1/2) = 1/4^(1/2) = 1/2. Product = 4 × (1/2) = 2.
What is log₃(81)?
Answer
4
We need y such that 3^y = 81 = 3^4. Therefore log₃(81) = 4. Alternatively, log₃(81) = log(81)/log(3) = 1.90849/0.47712 = 4.
Find the discriminant of 3x²−7x+5=0 and state the nature of the roots.
Answer
D = −11; two complex conjugate roots
D = b²−4ac = (−7)²−4(3)(5) = 49−60 = −11. Since D < 0, the equation has no real roots — the two roots are complex conjugates.
The 4th term of an AP is 13 and the 9th term is 33. Find the common difference d and the first term a₁.
Answer
d = 4; a₁ = 1
Using a_n=a₁+(n−1)d: a₄=a₁+3d=13 ... (1); a₉=a₁+8d=33 ... (2). Subtracting (1) from (2): 5d=20 → d=4. From (1): a₁=13−12=1.
Find the sum to infinity of the GP: 12, 4, 4/3, ...
Answer
S∞ = 18
Common ratio r = 4/12 = 1/3. Since |r|=1/3 < 1, the infinite sum exists. S∞ = a₁/(1−r) = 12/(1−1/3) = 12/(2/3) = 12 × (3/2) = 18.
Find the 4th term of (x + 2)^7.
Answer
C(7,3) · x^4 · 2^3 = 35 · x^4 · 8 = 280x^4
The (r+1)th term formula: T_(r+1)=C(n,r)·a^(n−r)·b^r. For the 4th term, r=3, n=7, a=x, b=2. T₄=C(7,3)·x^(7−3)·2^3 = 35·x⁴·8 = 280x⁴.
Worker A can finish a report in 8 hours, Worker B in 12 hours. Working together, how long will it take?
Answer
t = 24/5 = 4.8 hours
Rate_A = 1/8, Rate_B = 1/12. Combined rate = 1/8 + 1/12 = 3/24 + 2/24 = 5/24 job/hour. Time = 1/(5/24) = 24/5 = 4.8 hours.
How many liters of pure alcohol must be added to 40 L of a 20% alcohol solution to produce a 50% alcohol solution?
Answer
24 L
Let x = liters of pure alcohol (100% concentration) added. 0.20(40) + 1.00(x) = 0.50(40+x). 8 + x = 20 + 0.5x. 0.5x = 12. x = 24 L.
What value of k makes kx²−3x+3=0 have exactly one real solution?
Answer
k = 3/4
Exactly one real solution means a repeated root, so the discriminant D = 0. D = (−3)²−4(k)(3) = 9−12k = 0. Solving: 12k = 9 → k = 9/12 = 3/4.
Two cars leave the same point traveling in opposite directions. Car A at 80 km/h and Car B at 60 km/h. How long until they are 420 km apart?
Answer
t = 3 hours
Opposite directions: effective separation rate = 80+60 = 140 km/h. Time = Distance/Rate = 420/140 = 3 hours.
Diagrams
Algebra and Fundamentals — Complete Topic Overview
A hierarchical mind map showing all major topics in Algebra and Fundamentals and their sub-concepts. Use this to orient yourself before studying and as a final checklist before the board exam.
Quadratic Equation Solution Decision Tree
A decision flowchart guiding you through solving any quadratic equation: try factoring first, then compute the discriminant, interpret its sign to determine root nature, apply the quadratic formula, and verify using sum and product of roots.
Progression Type Identification and Formula Selection
A flowchart for identifying whether a given sequence is AP, GP, or HP, and then selecting the correct formula. This mirrors the thought process you should use on board exam day when faced with a progression problem.
Word Problem Type Identification and Setup Strategy
A comprehensive flowchart for identifying word problem type (Work, Mixture, Motion, or Age) and immediately setting up the correct equation. Following this process ensures you never mix up the setup strategies.
Approximate Distribution of Algebra Topics in PRC Board Exams (MSTE)
An approximate distribution of how frequently each algebra sub-topic appears in the PRC Civil Engineer board exams. Word problems and progressions dominate — prioritize these for maximum score impact.
Exam strategy
In the MSTE board exam, Algebra and Fundamentals items are among the most straightforward to score if you have your formulas memorized and your problem-setup process is disciplined. Follow this 4-step board exam protocol: (1) READ the problem completely before writing anything — identify the type (AP, GP, quadratic, work, mixture, motion). (2) WRITE the governing formula immediately (do not derive from scratch under exam pressure). (3) SUBSTITUTE carefully, watching signs, especially for negative bases, negative common differences, and drain-type work problems. (4) VERIFY using sum/product of roots, back-substitution, or a quick sanity check (e.g., combined time must be less than the faster individual time). Prioritize: work problems and progressions appear on every board exam — master these first. Then quadratic equations and logarithms. Binomial theorem is high-yield but narrower in scope. Allocate no more than 3 minutes per algebra item. If stuck after 2 minutes, use the answer choices: substitute each back into the original equation and identify which one satisfies it (reverse-engineering). In the week before the exam, practice 20 word problems per day without a calculator for the first pass — this builds the setup instinct that saves time on exam day.