ACT Math Hard Practice

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Algebra & Functions

Linear, Quadratic, Polynomial, Rational

Must Memorize

Quadratic Formula\(x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\)

DiscriminantIf \(b^2-4ac > 0\): 2 real roots
= 0: 1 root
< 0: no real roots

Vertex Form\(y = a(x-h)^2 + k\), vertex at \((h,k)\)

Sum/Product of RootsSum \(= -b/a\), Product \(= c/a\)

💡 Quick Example

If \(f(x) = 2x^2 - 8x + 6\), find the vertex.

→ Complete the square or use \(h = -b/2a = 8/4 = 2\), \(k = f(2) = 8-16+6 = -2\). Vertex: \((2, -2)\)

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Geometry & Trigonometry

Circles, Triangles, Coordinate Geometry

Must Memorize

Law of Cosines\(c^2 = a^2+b^2-2ab\cos C\)

SOHCAHTOAsin = opp/hyp
cos = adj/hyp
tan = opp/adj

CircleArea \(= \pi r^2\)
Circumference \(= 2\pi r\)

Pythagorean Triples3-4-5, 5-12-13, 8-15-17

Special Angles: sin 30° = ½ · cos 30° = √3/2 · tan 45° = 1 · sin 60° = √3/2

💡 Quick Example

A triangle has sides 7, 8 and included angle 60°. Find the third side.

→ \(c^2 = 49 + 64 - 2(7)(8)\cos 60° = 113 - 56 = 57\), so \(c = \sqrt{57} \approx 7.55\)

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Statistics & Probability

Mean, Median, Standard Deviation, Combinations

Must Memorize

Combinations\(\binom{n}{r} = \dfrac{n!}{r!(n-r)!}\)

Permutations\(P(n,r) = \dfrac{n!}{(n-r)!}\)

ProbabilityP(A or B) = P(A)+P(B)−P(A∩B)

Standard DeviationMeasures spread from mean; larger SD = more spread

💡 Quick Example

From 6 students, how many ways to select a team of 4?

→ \(\binom{6}{4} = \frac{6!}{4! \cdot 2!} = 15\) ways

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Number Theory & Pre-Calc

Sequences, Logarithms, Complex Numbers

Must Memorize

Arithmetic Seq.\(a_n = a_1 + (n-1)d\)
\(S_n = \frac{n}{2}(a_1 + a_n)\)

Geometric Seq.\(a_n = a_1 \cdot r^{n-1}\)
\(S_n = \frac{a_1(1-r^n)}{1-r}\)

Log Laws\(\log ab = \log a + \log b\)
\(\log(a/b) = \log a - \log b\)

Complex Numbers\(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\)

💡 Quick Example

Solve: \(\log_2(x+3) = 4\)

→ \(x + 3 = 2^4 = 16\), so \(x = 13\)

Answer Key & Full Solutions