Linear equations

Slope-intercept form

$y = mx + b$

Plot a line when you know its slope $m$ and y-intercept $b$ .

Point-slope form

$y - y_1 = m(x - x_1)$

Build a line from any one point and the slope.

Slope between two points

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Compute slope when two points are given.

Quadratic equations

Standard form

$ax^2 + bx + c = 0$

Recognise a quadratic; $a\neq 0$ .

Quadratic formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Always solves any quadratic — use when factoring is unclear.

Discriminant

$\Delta = b^2 - 4ac$

Tells you root nature: $\Delta>0$ two real, $=0$ repeated, $<0$ complex.

Sum and product of roots

$x_1 + x_2 = -\frac{b}{a},\quad x_1 x_2 = \frac{c}{a}$

Verify factoring without solving.

Exponents

Product of powers

$a^m \cdot a^n = a^{m+n}$

Same base — add the exponents.

Power of a power

$(a^m)^n = a^{mn}$

Multiply nested exponents.

Negative exponent

$a^{-n} = \frac{1}{a^n}$

Move base across the fraction bar to flip sign.

Zero exponent

$a^0 = 1\quad (a \neq 0)$

Anything to the zero is 1.

Logarithms

Definition

$\log_a b = c \iff a^c = b$

Logarithm is the inverse of exponentiation.

Product rule

$\log_a(xy) = \log_a x + \log_a y$

Convert product inside log to sum of logs.

Change of base

$\log_a b = \frac{\log_c b}{\log_c a}$

Compute any log using a calculator that only has $\log_{10}$ or $\ln$ .

Polynomial identities

Difference of squares

$a^2 - b^2 = (a-b)(a+b)$

Most common factoring shortcut — recognise instantly.

Perfect square

$(a \pm b)^2 = a^2 \pm 2ab + b^2$

Expand or recognise to factor in one step.

Sum/difference of cubes

$a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$

Less common but appears in standardised tests.

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