Solving Quadratic Equations

Identify and graph quadratic functions, solve quadratic equations graphically and numerically, and solve monic quadratic equations with integer roots algebraically.

A monic quadratic factorises into two brackets. Find two numbers that multiply to the constant and add to the middle term.

Builds onYr 9 · Graphing Quadratic Functions

Method

Rearrange to equal zero first

x^2 + bx + c = 0

Nothing works until one side is zero.

Find two numbers: multiply to $c$, add to $b$

This is the whole method for monic quadratics.

The null factor law

AB = 0 \Rightarrow A = 0 \text{ or } B = 0

If a product is zero, one factor must be zero.

Two brackets give two solutions

Both are valid unless the context rules one out.

Worked example

e.g.

\text{Solve } x^2 + 5x + 6 = 0

Check it equals zero. It does. $x^2 + 5x + 6 = 0$
Find two numbers that multiply to 6 and add to 5. $2 \times 3 = 6 \quad 2 + 3 = 5$
Write the factorised form. $(x + 2)(x + 3) = 0$
Apply the null factor law. $x + 2 = 0 \ \text{ or } \ x + 3 = 0$
Solve each bracket. $x = -2 \ \text{ or } \ x = -3$
Verify by substituting one back. $(-2)^2 + 5(-2) + 6 = 0 \quad\checkmark$

Practice

Rearrange to zero, factorise, then set each bracket to zero.

x^2 + 7x + 12 = 0

Answer

x = -3 \text{ or } -4

x^2 - 5x + 6 = 0

Answer

x = 2 \text{ or } 3

x^2 - 9 = 0

Answer

x = \pm 3

x^2 + 2x = 15

Answer

x = 3 \text{ or } -5

x^2 - 6x = 0

Answer

x = 0 \text{ or } 6

\text{Why must one side be zero first?}

Answer

\text{The null factor law only works against zero.}