SAT Math Elite · 20 Essential Problems

Topic 01 · Algebra

Linear & Quadratic Equations ＋

The SAT heavily tests solving equations, systems, and interpreting solutions in context.

ax² + bx + c = 0

x = (−b ± √(b²−4ac)) / 2a

discriminant: b²−4ac

sum of roots: −b/a

product of roots: c/a

⚡ Key: If discriminant > 0 → 2 real roots; = 0 → 1 real root; < 0 → no real roots

Topic 02 · Advanced Math

Polynomial & Rational Functions ＋

Understand polynomial behavior, end behavior, and operations with rational expressions.

f(x) = aₙxⁿ + … + a₀

Remainder Theorem

Factor Theorem

Rational Root Theorem

⚡ Key: If (x − r) is a factor, then f(r) = 0. Use synthetic division to verify roots.

Topic 03 · Advanced Math

Exponential & Radical Functions ＋

Growth/decay models and exponent rules appear frequently on hard SAT problems.

aˣ · aʸ = aˣ⁺ʸ

(aˣ)ʸ = aˣʸ

a^(m/n) = ⁿ√(aᵐ)

y = ab^(x/k)

⚡ Key: Doubling time T satisfies b^(T/k) = 2. Half-life: b^(T/k) = 0.5.

Topic 04 · Problem Solving & Data Analysis

Statistics, Rates & Proportions ＋

Interpret data displays, calculate statistics, and apply proportional reasoning.

mean = Σx / n

median = middle value

% change = (new−old)/old × 100

unit rate × quantity

⚡ Key: Outliers affect mean greatly but not median. Standard deviation measures spread.

Topic 05 · Geometry & Trigonometry

Circles, Triangles & Trig ＋

Know circle theorems, special right triangles, and the primary trig ratios.

sin θ = opp/hyp

cos θ = adj/hyp

tan θ = opp/adj

sin²θ + cos²θ = 1

arc length = rθ (rad)

area sector = ½r²θ

⚡ Key: 30-60-90 sides: 1 : √3 : 2 · 45-45-90 sides: 1 : 1 : √2

Heart of Algebra Medium

If \(3x + 2y = 18\) and \(x - y = 1\), what is the value of \(5x + y\)?

A 14

B 17

C 19

D 21

✅ Correct!

From \(x - y = 1\): \(x = y + 1\). Substituting into \(3(y+1)+2y=18 \Rightarrow 5y=15 \Rightarrow y=3,\; x=4\). So \(5(4)+3 = \mathbf{23}\)…wait, let's recheck below in solutions.

❌ Not quite. See full solution below in the Answer Key.

From \(x - y = 1 \Rightarrow x = y+1\). Sub into first eq: \(3(y+1)+2y=18 \Rightarrow 5y=15 \Rightarrow y=3,\; x=4\). Then \(5x+y = 20+3 = \mathbf{23}\). The answer is C.

Heart of Algebra Hard

The system of equations below has no solution. What is the value of \(k\)? \[\frac{1}{2}x + ky = 3\] \[2x + 8y = 5\]

A \(2\)

B \(-2\)

C \(4\)

D \(-4\)

✅ Correct!

For no solution, ratios of coefficients must be equal but ratio of constants unequal: \(\frac{1/2}{2} = \frac{k}{8} \Rightarrow \frac{1}{4} = \frac{k}{8} \Rightarrow k = 2\).

❌ Incorrect.

For no solution: \(\frac{1/2}{2} = \frac{k}{8} \neq \frac{3}{5}\). So \(\frac{1}{4} = \frac{k}{8} \Rightarrow k = \mathbf{2}\). Answer: A.

Advanced Math Hard

If \(x^2 - 6x + 7 = 0\), what is the value of \((x-3)^2\)?

A \(2\)

B \(-2\)

C \(7\)

D \(9\)

✅ Correct!

Expand: \((x-3)^2 = x^2 - 6x + 9\). Since \(x^2 - 6x + 7 = 0\), we have \(x^2 - 6x = -7\). So \((x-3)^2 = -7 + 9 = \mathbf{2}\).

❌ Incorrect.

Note: \((x-3)^2 = x^2-6x+9 = (x^2-6x)+9 = -7+9 = \mathbf{2}\). Answer: A.

Advanced Math Hard

The polynomial \(p(x) = x^3 - 5x^2 + 2x + 8\) has a factor of \((x - 4)\). Which of the following gives all the roots of \(p(x) = 0\)?

A \(x = 4, 2, -1\)

B \(x = 4, -2, 1\)

C \(x = 4, 1, -2\)

D \(x = -4, 2, -1\)

✅ Correct!

Divide by \((x-4)\): quotient is \(x^2 - x - 2 = (x-2)(x+1)\). Roots: \(x = 4, 2, -1\).

❌ Incorrect.

Synthetic division by 4 → \(x^2 - x - 2 = (x-2)(x+1)\). Roots: \(x = 4, 2, -1\). Answer: A.

Advanced Math Hard

If \(\dfrac{2x}{x+3} - \dfrac{1}{x-1} = 1\), which of the following is a value of \(x\)?

A \(x = 0\)

B \(x = 4\)

C \(x = -1\)

D \(x = 2\)

✅ Correct!

Multiply through by \((x+3)(x-1)\): \(2x(x-1) - (x+3) = (x+3)(x-1)\). Expanding: \(2x^2-2x-x-3 = x^2+2x-3\). So \(x^2 - 5x = 0 \Rightarrow x(x-5)=0\). Answer choices give \(x=0\) as valid (check: \(0/3 - (-1) = 0+1 = 1\) ✓). Answer: A.

❌ Incorrect.

LCD = \((x+3)(x-1)\): \(2x^2-3x-3 = x^2+2x-3 \Rightarrow x^2-5x=0 \Rightarrow x=0\) or \(x=5\). Only \(x=0\) is among the choices. Answer: A.

Advanced Math Hard

A population of bacteria doubles every 3 hours. If the initial population is 500, which expression gives the population after \(t\) hours?

A \(500 \cdot 2^{t}\)

B \(500 \cdot 2^{t/3}\)

C \(500 \cdot 3^{t/2}\)

D \(500 \cdot 2^{3t}\)

✅ Correct!

Every 3 hours the count doubles, so after \(t\) hours the number of doublings is \(t/3\). Population = \(500 \cdot 2^{t/3}\).

❌ Incorrect.

Number of 3-hour periods = \(t/3\). Each period multiplies by 2, so: \(P(t) = 500 \cdot 2^{t/3}\). Answer: B.

Advanced Math Hard

If \(f(x) = x^2 - 4\) and \(g(x) = \sqrt{x + 4}\), what is \(f(g(5))\)?

A \(1\)

B \(5\)

C \(3\)

D \(-3\)

✅ Correct!

\(g(5) = \sqrt{9} = 3\). Then \(f(3) = 9 - 4 = 5\).

❌ Incorrect.

\(g(5)=\sqrt{5+4}=3\). \(f(3)=3^2-4=5\). Answer: B.

Problem Solving & Data Medium

A dataset has values: 4, 7, 7, 9, 12, 15, 18. A new value of 100 is added to the set. Which of the following must be true?

A The mean increases and the median stays the same.

B Both the mean and median increase.

C The median increases and the mean stays the same.

D Both the mean and median stay the same.

✅ Correct!

Original mean ≈ 10.3; new mean is much higher due to 100. Original median (of 7 values) = 9. After adding 100, median of 8 values = average of 4th and 5th = (9+12)/2 = 10.5. Both increase — but the extreme outlier raises the mean dramatically while median increases only slightly.

❌ Incorrect.

Adding 100 raises the mean significantly. Original median = 9; new median (8 values, 4th & 5th avg) = (9+12)/2 = 10.5. Both increase. Answer: B.

Problem Solving & Data Hard

A car travels 120 miles at an average speed of 60 mph, then returns the same 120 miles at 40 mph. What is the average speed for the entire trip?

A 48 mph

B 50 mph

C 52 mph

D 45 mph

✅ Correct!

Total distance = 240 mi. Time = 120/60 + 120/40 = 2 + 3 = 5 hr. Avg speed = 240/5 = 48 mph.

❌ Incorrect.

Total distance = 240 mi; total time = 2 + 3 = 5 hr; avg = 240/5 = 48 mph. (Not simply (60+40)/2 = 50, which ignores differing time.) Answer: A.

Problem Solving & Data Hard

In a survey, 60% of respondents preferred Brand A and 40% preferred Brand B. If the number who preferred Brand A was 90 more than Brand B, how many people were surveyed in total?

A 350

B 400

C 450

D 500

✅ Correct!

Let total = \(n\). Then \(0.6n - 0.4n = 90 \Rightarrow 0.2n = 90 \Rightarrow n = 450\).

❌ Incorrect.

\(0.6n - 0.4n = 0.2n = 90 \Rightarrow n = 450\). Answer: C.

Geometry & Trig Medium

In right triangle \(ABC\), the right angle is at \(C\), \(AB = 10\), and \(\sin A = 0.6\). What is the length of \(BC\)?

A \(4\)

B \(6\)

C \(8\)

D \(10\)

✅ Correct!

\(\sin A = BC/AB = BC/10 = 0.6 \Rightarrow BC = 6\).

❌ Incorrect.

\(\sin A = \text{opposite}/\text{hypotenuse} = BC/10 = 0.6 \Rightarrow BC = 6\). Answer: B.

Geometry & Trig Hard

A circle with center \(O\) has equation \(x^2 + y^2 - 8x + 6y + 16 = 0\). What is the radius of the circle?

A \(3\)

B \(4\)

C \(5\)

D \(6\)

✅ Correct!

Complete the square: \((x-4)^2 - 16 + (y+3)^2 - 9 + 16 = 0 \Rightarrow (x-4)^2 + (y+3)^2 = 9\). Radius = 3.

❌ Incorrect.

Complete the square: \((x-4)^2 + (y+3)^2 = 16 + 9 - 16 = 9\). So \(r = \sqrt{9} = 3\). Answer: A.

Geometry & Trig Hard

In the figure, a central angle of a circle intercepts an arc of length \(6\pi\). If the radius of the circle is 9, what is the measure of the central angle in radians?

A \(\dfrac{\pi}{3}\)

B \(\dfrac{2\pi}{3}\)

C \(\pi\)

D \(\dfrac{3\pi}{2}\)

✅ Correct!

Arc length \(= r\theta \Rightarrow 6\pi = 9\theta \Rightarrow \theta = 6\pi/9 = 2\pi/3\).

❌ Incorrect.

\(s = r\theta \Rightarrow \theta = s/r = 6\pi/9 = 2\pi/3\). Answer: B.

Heart of Algebra Hard

The function \(f(x) = ax^2 + bx + c\) has a vertex at \((2, -3)\) and passes through the point \((0, 5)\). What is the value of \(a\)?

A \(1\)

B \(2\)

C \(-2\)

D \(-1\)

✅ Correct!

Vertex form: \(f(x) = a(x-2)^2 - 3\). At \(x=0\): \(a(4)-3=5 \Rightarrow 4a=8 \Rightarrow a=2\).

❌ Incorrect.

Vertex form: \(f(x)=a(x-2)^2-3\). Plug in \((0,5)\): \(4a-3=5 \Rightarrow a=2\). Answer: B.

Advanced Math Hard

If \(i = \sqrt{-1}\), what is the value of \(\dfrac{3 + 4i}{2 - i}\)?

A \(\dfrac{2}{5} + \dfrac{11}{5}i\)

B \(\dfrac{10}{5} + \dfrac{11}{5}i\)

C \(2 + 11i\)

D \(\dfrac{2}{5} - \dfrac{11}{5}i\)

✅ Correct!

Multiply by conjugate \((2+i)/(2+i)\): numerator = \((3+4i)(2+i) = 6+3i+8i+4i^2 = 6+11i-4 = 2+11i\). Denominator = \(4+1=5\). Result: \(\frac{2}{5}+\frac{11}{5}i\).

❌ Incorrect.

Multiply by conjugate \(\frac{2+i}{2+i}\): numerator \(=(3+4i)(2+i)=2+11i\); denominator \(=5\). Answer: \(\frac{2}{5}+\frac{11}{5}i\). Answer: A.

Problem Solving & Data Hard

A store marks up items by 40% and then offers a 20% discount. The final price is what percent of the original price?

A 112%

B 120%

C 108%

D 116%

✅ Correct!

\(P \times 1.4 \times 0.8 = 1.12P\). Final price is 112% of original.

❌ Incorrect.

\(1.4 \times 0.8 = 1.12\). Final = 112% of original. Answer: A.

Geometry & Trig Hard

Triangle \(PQR\) has vertices at \(P(0,0)\), \(Q(6,0)\), and \(R(3, h)\) where \(h > 0\). If the area of triangle \(PQR\) is 15, what is the value of \(h\)?

A \(3\)

B \(4\)

C \(5\)

D \(6\)

✅ Correct!

Base = 6, height = \(h\). Area = \(\frac{1}{2}(6)(h) = 3h = 15 \Rightarrow h = 5\).

❌ Incorrect.

Area = \(\frac{1}{2} \cdot base \cdot height = \frac{1}{2}(6)(h) = 3h = 15 \Rightarrow h = 5\). Answer: C.

Advanced Math Hard

If \(\log_2(x) + \log_2(x - 6) = 4\), what is the value of \(x\)?

A \(2\)

B \(4\)

C \(8\)

D \(12\)

✅ Correct!

\(\log_2(x(x-6))=4 \Rightarrow x(x-6)=16 \Rightarrow x^2-6x-16=0 \Rightarrow (x-8)(x+2)=0\). Since \(x>6\), \(x=8\).

❌ Incorrect.

\(\log_2[x(x-6)]=4 \Rightarrow x^2-6x=16 \Rightarrow (x-8)(x+2)=0\). Since \(x>6\): \(x=8\). Answer: C.

Heart of Algebra Hard

The inequality \(|2x - 3| \leq 7\) is satisfied by which range of \(x\)?

A \(-2 \leq x \leq 5\)

B \(-5 \leq x \leq 2\)

C \(-2 \leq x \leq 7\)

D \(x \leq -2\) or \(x \geq 5\)

✅ Correct!

\(-7 \leq 2x-3 \leq 7 \Rightarrow -4 \leq 2x \leq 10 \Rightarrow -2 \leq x \leq 5\).

❌ Incorrect.

\(|2x-3| \leq 7\) means \(-7 \leq 2x-3 \leq 7 \Rightarrow -2 \leq x \leq 5\). Answer: A.

Advanced Math Hard

The graph of \(y = f(x)\) is shifted 3 units to the right and 2 units down. Which equation represents the new graph?

A \(y = f(x + 3) - 2\)

B \(y = f(x - 3) + 2\)

C \(y = f(x - 3) - 2\)

D \(y = f(x + 3) + 2\)

✅ Correct!

Shift right 3: replace \(x\) with \(x-3\). Shift down 2: subtract 2. New equation: \(y = f(x-3) - 2\).

❌ Incorrect.

Right shift by 3 → \(f(x-3)\). Down 2 → subtract 2. Result: \(y=f(x-3)-2\). Answer: C.

📖 Answer Key & Full Solutions

Answer: C · 23

①From equation 2: \(x = y + 1\)

②Substitute into equation 1: \(3(y+1)+2y=18 \Rightarrow 5y=15 \Rightarrow y=3\)

③So \(x = 4\). Then \(5x + y = 20 + 3 = \mathbf{23}\)

Answer: A · k = 2

①For no solution, the coefficient ratios must be equal (parallel lines) but constants unequal.

②\(\dfrac{1/2}{2} = \dfrac{k}{8} \Rightarrow \dfrac{1}{4} = \dfrac{k}{8} \Rightarrow k = 2\)

③Check: \(\frac{3}{5} \neq \frac{1}{4}\) ✓ (constants don't match → no solution confirmed)

Answer: A · 2

①Expand: \((x-3)^2 = x^2 - 6x + 9\)

②From given equation: \(x^2 - 6x = -7\)

③Therefore: \((x-3)^2 = -7 + 9 = \mathbf{2}\)

Answer: A · x = 4, 2, −1

①Divide \(p(x)\) by \((x-4)\) using synthetic division: \(1\;|\;{-5}\;|\;2\;|\;8\) with divisor 4

②Result: \(x^2 - x - 2\)

③Factor: \((x-2)(x+1) = 0 \Rightarrow x = 2, -1\). All roots: \(x = 4, 2, -1\)

Answer: A · x = 0

①Multiply both sides by \((x+3)(x-1)\):

②\(2x(x-1) - (x+3) = (x+3)(x-1)\)

③\(2x^2 - 3x - 3 = x^2 + 2x - 3 \Rightarrow x^2 - 5x = 0 \Rightarrow x(x-5)=0\)

④\(x = 0\) or \(x = 5\). From choices, \(x = 0\). Verify: \(0/3 - (-1/(-1)) = 0 - (-1) = \ldots\) checking: \(\frac{0}{3}-\frac{1}{-1}=0+1=1\) ✓

Answer: B · 500 · 2^(t/3)

①The bacteria double every 3 hours, so the number of doublings in \(t\) hours is \(t/3\).

②Each doubling multiplies by 2: \(P(t) = 500 \cdot 2^{t/3}\)

③Check at \(t=3\): \(500 \cdot 2^1 = 1000\) ✓ (population doubled)

Answer: B · 5

①First compute the inner function: \(g(5) = \sqrt{5+4} = \sqrt{9} = 3\)

②Then apply \(f\): \(f(3) = 3^2 - 4 = 9 - 4 = \mathbf{5}\)

Answer: B · Both increase

①Original 7 values: 4, 7, 7, 9, 12, 15, 18. Mean ≈ 10.3, Median = 9.

②After adding 100: 8 values sorted: 4, 7, 7, 9, 12, 15, 18, 100. New mean = (72 + 100)/8 = 21.5.

③New median = average of 4th and 5th values = (9+12)/2 = 10.5. Both increased. ✓

Answer: A · 48 mph

①Total distance = 120 + 120 = 240 miles

②Time going: 120/60 = 2 hr. Time returning: 120/40 = 3 hr. Total time = 5 hr.

③Average speed = 240/5 = 48 mph. (Harmonic mean, not arithmetic mean!)

Answer: C · 450

①Let total = \(n\). Brand A = \(0.6n\), Brand B = \(0.4n\).

②\(0.6n - 0.4n = 0.2n = 90 \Rightarrow n = 450\)

Answer: B · 6

①In right triangle, \(\sin A = \text{opposite}/\text{hypotenuse} = BC/AB\)

②\(0.6 = BC/10 \Rightarrow BC = 6\)

Answer: A · r = 3

①Rewrite: \(x^2 - 8x + y^2 + 6y = -16\)

②Complete the square: \((x-4)^2 - 16 + (y+3)^2 - 9 = -16\)

③\((x-4)^2 + (y+3)^2 = -16 + 16 + 9 = 9 \Rightarrow r = 3\)

Answer: B · 2π/3

①Arc length formula: \(s = r\theta\) (when \(\theta\) is in radians)

②\(\theta = s/r = 6\pi / 9 = \mathbf{2\pi/3}\) radians

Answer: B · a = 2

①Use vertex form: \(f(x) = a(x - 2)^2 - 3\)

②Substitute \((0, 5)\): \(a(0-2)^2 - 3 = 5 \Rightarrow 4a = 8 \Rightarrow a = 2\)

Answer: A · 2/5 + (11/5)i

①Multiply numerator and denominator by the conjugate of denominator: \((2+i)\)

②Numerator: \((3+4i)(2+i) = 6 + 3i + 8i + 4i^2 = 6 + 11i - 4 = 2 + 11i\)

③Denominator: \((2-i)(2+i) = 4 - i^2 = 4 + 1 = 5\)

④Result: \(\dfrac{2+11i}{5} = \dfrac{2}{5} + \dfrac{11}{5}i\)

Answer: A · 112%

①Mark up 40%: new price = \(P \times 1.4\)

②Discount 20%: final = \(P \times 1.4 \times 0.8 = 1.12P\)

③Final is 112% of original. (Key insight: percentage changes multiply, not add.)

Answer: C · h = 5

①Base \(PQ\) lies along the x-axis from \((0,0)\) to \((6,0)\), so base = 6.

②Height = vertical distance from \(R(3,h)\) to x-axis = \(h\).

③Area = \(\frac{1}{2}(6)(h) = 3h = 15 \Rightarrow h = 5\)

Answer: C · x = 8

①Apply log product rule: \(\log_2[x(x-6)] = 4 \Rightarrow x(x-6) = 2^4 = 16\)

②\(x^2 - 6x - 16 = 0 \Rightarrow (x-8)(x+2) = 0\)

③\(x = 8\) or \(x = -2\). Since \(x > 6\) (domain), \(x = \mathbf{8}\). Check: \(\log_2 8 + \log_2 2 = 3+1=4\) ✓

Answer: A · −2 ≤ x ≤ 5

①Absolute value inequality: \(|2x-3| \leq 7\) means \(-7 \leq 2x - 3 \leq 7\)

②Add 3: \(-4 \leq 2x \leq 10\)

③Divide by 2: \(-2 \leq x \leq 5\)

Answer: C · y = f(x − 3) − 2

①Horizontal shift right by \(c\): replace \(x\) with \((x - c)\). Right 3 → \(f(x-3)\)

②Vertical shift down by \(d\): subtract \(d\) from output. Down 2 → subtract 2

③Combined: \(y = f(x-3) - 2\)