Solving Quadratic Equations – Step by Step

A guided lesson on solving quadratic equations using factoring, completing the square, and the quadratic formula.

What is a Quadratic Equation?

A quadratic equation has degree 2:

$ax^2 + bx + c = 0$

There are three main methods to solve them.

Method 1: Factoring

If the quadratic factors nicely:

$x^2 - 5x + 6 = (x-2)(x-3) = 0$

So $x = 2$ or $x = 3$ .

1
Write in standard form
$ax^2 + bx + c = 0$
2
Find two numbers that multiply to c and add to b
For $x^2 - 5x + 6$ : need $p \cdot q = 6$ and $p + q = -5$ . So $p = -2, q = -3$ .
3
Factor and solve
$(x - 2)(x - 3) = 0 \implies x = 2$ or $x = 3$ .

Method 2: The Quadratic Formula

Always works:

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

Solve x² + 2x − 8 = 0 using the formula

$a=1, b=2, c=-8$

$\Delta = 4 + 32 = 36$

$x = \frac{-2 \pm 6}{2}$ : $x_1 = 2$ , $x_2 = -4$

Method 3: Completing the Square

Useful for understanding the vertex form $a(x-h)^2 + k$ .

1
Move constant to the right
$x^2 + 4x = 5$
2
Add (b/2)² to both sides
$x^2 + 4x + 4 = 5 + 4 = 9$
3
Factor and solve
$(x+2)^2 = 9 \implies x + 2 = \pm 3 \implies x = 1$ or $x = -5$

Tip

If the discriminant $b^2 - 4ac < 0$ , the equation has no real solutions.

Real world: ball thrown into the air

A ball is launched upward with the height (in metres) modelled by $h = -5t^2 + 20t$ , where $t$ is time in seconds. Setting $h = 0$ to find when it lands: $-5t(t - 4) = 0$ , so $t = 0$ or $t = 4$ seconds. Quadratic equations describe every projectile — from basketballs to rockets.

Real world: satellite dish shape

A parabolic satellite dish has a cross-section described by $y = 0.1x^2$ . To find where the dish is 5 m wide at height $y = 0.9$ m: $0.9 = 0.1x^2 \Rightarrow x^2 = 9 \Rightarrow x = \pm 3$ m. Engineers use quadratic equations to design the exact curvature that focuses signals onto the receiver.

Real world: fencing a rectangular garden

A gardener has 40 m of fence and wants to enclose a rectangular area of 96 m². Setting up: $2l + 2w = 40$ so $w = 20 - l$ , then $l(20-l) = 96 \Rightarrow l^2 - 20l + 96 = 0 \Rightarrow (l-8)(l-12) = 0$ . The garden is $8\text{ m} \times 12\text{ m}$ .

Tip Before choosing a method, check the discriminant

b^2 - 4ac

. If it is a perfect square, factoring will be quick. Otherwise use the quadratic formula.

Practice Problems

Factor: x² − 7x + 12 = 0

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(x−3)(x−4) = 0 → x = 3 or x = 4

Use quadratic formula: x² − 4x + 3 = 0

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Δ = 16−12 = 4. x = (4±2)/2 → x = 3 or x = 1

Complete the square: x² + 6x − 7 = 0

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(x+3)² = 16 → x+3 = ±4 → x = 1 or x = −7