The Quadratic Formula

The quadratic formula solves any quadratic equation ax² + bx + c = 0 by finding the roots in terms of a, b, and c.

The Quadratic Equation

A quadratic equation has the form:

$ax^2 + bx + c = 0, \quad a \neq 0$

Quadratic Formula

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

This formula gives both roots of any quadratic equation.

The Discriminant

The discriminant $\Delta = b^2 - 4ac$ determines the nature of the roots:

Discriminant	Roots
$\Delta > 0$	Two distinct real roots
$\Delta = 0$	One repeated real root
$\Delta < 0$	Two complex conjugate roots

Note

The sum of roots is $x_1 + x_2 = -b/a$ and the product is $x_1 \cdot x_2 = c/a$ (Vieta's formulas).

Solve x² − 5x + 6 = 0

Here $a = 1$ , $b = -5$ , $c = 6$ .

$\Delta = (-5)^2 - 4(1)(6) = 25 - 24 = 1$

$x = \dfrac{5 \pm 1}{2}$

So $x_1 = 3$ and $x_2 = 2$ .

Completing the Square

Another method: rewrite $ax^2 + bx + c$ as $a(x + h)^2 + k$ .

1
Divide by a
$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$
2
Move constant to right side
$x^2 + \frac{b}{a}x = -\frac{c}{a}$
3
Add (b/2a)² to both sides
$\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$
4
Take the square root
$x + \frac{b}{2a} = \pm\frac{\sqrt{b^2 - 4ac}}{2a}$

Real-World Examples

Real world: when does a ball hit the ground?

A ball is launched upward. Its height in metres is $h = -5t^2 + 20t$ .

To find when it lands, solve $-5t^2 + 20t = 0$ , or equivalently $t^2 - 4t = 0$ .

Using the quadratic formula: $\Delta = 16 - 0 = 16$ , so $t = \frac{4 \pm 4}{2}$ .

$t = 4$ s (lands) and $t = 0$ (launch). The ball is in the air for 4 seconds.

Real world: rectangular garden dimensions

A gardener wants a rectangular plot with area 24 m² and perimeter 20 m. Let one side be $x$ .

From the perimeter: the other side is $10 - x$ . Area gives $x(10 - x) = 24$ , so $x^2 - 10x + 24 = 0$ .

$\Delta = 100 - 96 = 4$ , so $x = \frac{10 \pm 2}{2}$ : sides are 6 m and 4 m.

Real world: stopping distance

A car has stopping distance $d = 0.05v^2 + 0.7v$ . A driver needs to stop within 40 m.

Solve $0.05v^2 + 0.7v - 40 = 0$ , multiply by 20: $v^2 + 14v - 800 = 0$ .

$\Delta = 196 + 3200 = 3396$ , so $v \approx \frac{-14 + 58.3}{2} \approx 22$ m/s (≈ 79 km/h). Driving faster is unsafe for that stopping distance.

Practice Problems

Solve x² + 4x − 5 = 0 using the quadratic formula

Show answer

Δ = 16 + 20 = 36. x = (−4 ± 6)/2. x₁ = 1, x₂ = −5

Find the discriminant of 2x² − 3x + 5 = 0. How many real roots?

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Δ = 9 − 40 = −31 < 0. No real roots (two complex roots).

Solve x² − 6x + 9 = 0

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Δ = 36 − 36 = 0. One repeated root: x = 3