Math Mastery — Algebra 2 & Geometry

Algebra 2 10 problems

Quadratic Functions

Which of the following is the vertex form of \(f(x) = x^2 - 6x + 5\)?

💡VERTEX FORM: complete the square → \((x-h)^2+k\)

Polynomial Operations

What is the remainder when \(p(x) = 2x^3 - 3x^2 + x - 5\) is divided by \((x - 2)\)?

💡REMAINDER THEOREM: plug in the zero → \(p(2) =\) remainder

Rational Exponents

Simplify: \(27^{2/3}\)

💡RATIONAL EXPONENT: denominator = root, numerator = power → \(\left(\sqrt[3]{27}\right)^2\)

Logarithms

Solve for \(x\): \(\log_3(x+2) + \log_3(x-2) = 3\)

💡LOG PRODUCT: \(\log A + \log B = \log(AB)\), then convert to exponential

Complex Numbers

What is \((3 + 2i)(1 - 4i)\) in the form \(a + bi\)?

💡FOIL + REMEMBER: \(i^2 = -1\), replace and combine real/imaginary parts

Systems of Equations

How many solutions does this system have? \[\begin{cases} y = x^2 - 4 \\ y = 2x - 1 \end{cases}\]

💡SUBSTITUTE & DISCRIMINANT: set equal → \(\Delta = b^2 - 4ac\). \(\Delta > 0\): 2 solutions

Sequences & Series

Find the sum of the first 8 terms of the geometric sequence \(3, 6, 12, 24, \ldots\)

💡GEO SUM: \(S_n = a_1 \cdot \dfrac{r^n - 1}{r - 1}\), identify \(r\) first

Inverse Functions

If \(f(x) = \dfrac{2x+1}{x-3}\), what is \(f^{-1}(x)\)?

💡INVERSE SWAP: swap \(x\) & \(y\), then isolate \(y\)

Conic Sections — Parabola

The equation \(x^2 - 4x - 8y + 4 = 0\) represents a parabola. What is the vertex?

💡STANDARD PARABOLA: complete the square in \(x\) → \((x-h)^2 = 4p(y-k)\), vertex \(= (h,k)\)

Matrices

If \(A = \begin{pmatrix}2 & 1\\3 & 4\end{pmatrix}\), what is \(\det(A)\) and does \(A^{-1}\) exist?

💡DETERMINANT: \(ad - bc\). If \(\det \neq 0\) → inverse exists

Geometry 10 problems

Triangle Congruence

Two triangles have two pairs of equal sides and the included angle equal. Which congruence rule applies?

💡CONGRUENCE: SSS · SAS · ASA · AAS · HL — "included" = between the two sides

Circle Theorems

An inscribed angle intercepts an arc of \(140°\). What is the measure of the inscribed angle?

💡INSCRIBED ANGLE: = half the intercepted arc

Similarity & Scale

Two similar triangles have a side-length ratio of \(2:5\). What is the ratio of their areas?

💡SCALE FACTOR: sides → \(k\), areas → \(k^2\), volumes → \(k^3\)

Right Triangles — Pythagorean Theorem

In a right triangle, the two legs are \(5\) and \(12\). What is the hypotenuse?

💡PYTHAGOREAN: \(a^2 + b^2 = c^2\). Memorise: (3,4,5) (5,12,13) (8,15,17)

Coordinate Geometry

What is the equation of the perpendicular bisector of the segment with endpoints \(A(2, 4)\) and \(B(6, 8)\)?

💡PERP BISECTOR: midpoint + slope = \(-1/m\) of original segment

Parallel Lines & Transversals

Two parallel lines are cut by a transversal. One co-interior angle (same-side interior) is \(3x + 10\) and the other is \(5x - 30\). Find \(x\).

💡CO-INTERIOR: add to \(180°\) (supplementary), NOT equal

Volume — Solids

A cone and a cylinder share the same base radius \(r = 4\) and height \(h = 9\). What is the ratio of the cone's volume to the cylinder's volume?

💡CONE vs CYLINDER: cone = \(\frac{1}{3}\pi r^2 h\), cylinder = \(\pi r^2 h\) → ratio always \(1:3\)

Trigonometry — Right Triangles

In a right triangle, \(\sin\theta = \dfrac{3}{5}\). What is \(\cos\theta\)?

💡SOH-CAH-TOA + PYTHAGOREAN ID: \(\sin^2\theta + \cos^2\theta = 1\)

Quadrilaterals

The diagonals of a parallelogram bisect each other. If one diagonal has endpoints at \((1, 3)\) and \((7, 11)\), what is the midpoint of the other diagonal?

💡PARALLELOGRAM: diagonals bisect each other → share the same midpoint

Transformations

Point \(P(3, -2)\) is rotated \(90°\) counter-clockwise about the origin. What are the new coordinates?

💡ROTATION 90° CCW: \((x, y) \to (-y, x)\). CW: \((x, y) \to (y, -x)\)