Year 10AC9M10A01

The Quadratic Formula

Expand, factorise and simplify expressions and solve equations algebraically, applying exponent laws involving products, quotients and powers of variables, and the distributive property.

Things you should know.

Method

The quadratic formula

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

Works for every quadratic, factorising or not.

The discriminant

\Delta = b^{2} - 4ac

$\Delta > 0$: two roots. $\Delta = 0$: one root. $\Delta < 0$: no real roots.

Worked example

e.g.

2x^{2} + 5x - 3 = 0

Write in the form $ax^{2} + bx + c = 0$ and list $a$ , $b$ , $c$ $a = 2,\ b = 5,\ c = -3$
Calculate the discriminant first $\Delta = 5^{2} - 4(2)(-3) = 49$
Substitute into the formula $x = \frac{-5 \pm \sqrt{49}}{2(2)}$
Simplify the root $x = \frac{-5 \pm 7}{4}$
Split into two answers $x = \frac{2}{4} \ \text{ or } \ x = \frac{-12}{4}$
Simplify $x = \tfrac{1}{2} \ \text{ or } \ x = -3$

Practice

Solve using the quadratic formula:

x^{2} + 5x + 6 = 0

Answer

x = -2 \text{ or } x = -3

x^{2} - 7x + 12 = 0

Answer

x = 3 \text{ or } x = 4

2x^{2} + 7x + 3 = 0

Answer

x = -\tfrac{1}{2} \text{ or } x = -3

3x^{2} - 5x - 2 = 0

Answer

x = 2 \text{ or } x = -\tfrac{1}{3}

x^{2} - 4x + 4 = 0

Answer

x = 2 \ (\Delta = 0, \text{ one root})

x^{2} + 2x + 5 = 0

Answer

\Delta = -16 < 0 \ \therefore \text{ no real roots}

x^{2} - 2x - 1 = 0

Answer

x = 1 \pm \sqrt{2}

2x^{2} - 3x - 2 = 0

Answer

x = 2 \text{ or } x = -\tfrac{1}{2}

x^{2} - 6x + 7 = 0

Answer

x = 3 \pm \sqrt{2}

5x^{2} + 3x - 2 = 0

Answer

x = \tfrac{2}{5} \text{ or } x = -1