The quadratic formula

Some quadratics don't factor nicely, like $x^{2} + 4 x + 1 = 0$ . The formula works every time. It comes from completing the square on $a x^{2} + b x + c = 0$ , so it's not magic.

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

$b^{2} - 4 a c$	Real solutions	Graph
Positive	Two	Crosses the $x$ -axis twice
Zero	One	Touches it once
Negative	None	Never touches it

Work through it

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

Read the values, with their signs. $a = 2, b = 3, c = - 2$

Discriminant first. $9 - 4 (2) (- 2) = 25$

Formula. $x = \frac{- 3 \pm 5}{4}$

Two answers. $x = \frac{1}{2}$ or $x = - 2$

$x^{2} + 4 x + 1 = 0$

Discriminant. $16 - 4 = 12$

Simplify the root. $12 = 23$

Formula. $x = \frac{- 4 \pm 2 3}{2} = - 2 \pm 3$

Watch out for this

With $b = - 3$ , writing $- b$ as $- 3$ and $b^{2}$ as $- 9$ .

$- b$ means the opposite of $b$ , so 3. And $(- 3)^{2} = 9$ . Put negatives in parentheses.

The short version
Get

a x^{2} + b x + c = 0

first. Then

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

. The discriminant,

b^{2} - 4 a c

, tells you how many real solutions there are.

Your turn

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