Quadratic Formula: How to Solve Quadratic Equations with Solved Examples

Quadratic equations appear everywhere in school and college maths – and in physics, engineering and finance. This guide explains the quadratic formula, how to use the discriminant to predict the type of roots, and three methods to solve quadratic equations – factorisation, completing the square and the formula – with fully solved examples, including equations with complex roots and a real-life application.

What Is a Quadratic Equation?

A quadratic equation is an equation of degree 2 – the highest power of the variable is 2. Its standard form is:

ax² + bx + c = 0, where a ≠ 0

Here a, b and c are numbers called coefficients. If a were 0, the x² term would disappear and the equation would be linear. Examples: x² − 5x + 6 = 0 (a = 1, b = −5, c = 6) and 2x² + 3x − 2 = 0 (a = 2, b = 3, c = −2). Before solving, always rearrange the equation so that one side is zero.

The Quadratic Formula

x = [−b ± √(b² − 4ac)] ÷ 2a

The quadratic formula works for every quadratic equation, which is why it is the method to fall back on when an equation does not factorise easily. The "±" means there are usually two solutions, called roots: one using + and one using −.

The Discriminant: What Kind of Roots?

The expression under the square root, D = b² − 4ac, is called the discriminant. It tells you the nature of roots before you solve:

If D is a perfect square (such as 1, 4, 9, 25), the roots are rational and the equation can be factorised.

Solved Example 1: Two Real Roots

Solve 2x² + 3x − 2 = 0.

1. a = 2, b = 3, c = −2
2. D = 3² − 4 × 2 × (−2) = 9 + 16 = 25
3. √D = 5
4. x = (−3 + 5) ÷ 4 = 2 ÷ 4 = 0.5
5. x = (−3 − 5) ÷ 4 = −8 ÷ 4 = −2

Check: 2(0.5)² + 3(0.5) − 2 = 0.5 + 1.5 − 2 = 0 ✓

Solved Example 2: Equal Roots

Solve x² + 4x + 4 = 0. D = 16 − 16 = 0, so there is one repeated root: x = −4 ÷ 2 = −2. Indeed, x² + 4x + 4 = (x + 2)².

Solved Example 3: Complex Roots

Solve x² + 2x + 5 = 0. D = 4 − 20 = −16. Since D is negative there are no real roots. Using i = √−1, √−16 = 4i, so x = (−2 ± 4i) ÷ 2 = −1 ± 2i. Complex roots always come in conjugate pairs like this when a, b and c are real.

Method 1: Solving by Factorisation

Factorisation is the quickest method when it works. For x² − 5x + 6 = 0, find two numbers that multiply to 6 (the constant) and add to −5 (the x coefficient): −2 and −3.

x² − 5x + 6 = (x − 2)(x − 3) = 0

So x = 2 or x = 3. When a ≠ 1, use the "split the middle term" method: for 2x² + 3x − 2, find numbers that multiply to a × c = −4 and add to 3: 4 and −1. Then 2x² + 4x − x − 2 = 2x(x + 2) − 1(x + 2) = (2x − 1)(x + 2), giving x = 0.5 or x = −2.

Method 2: Completing the Square

Completing the square rewrites the equation as a perfect square. It is also how the quadratic formula is derived.

Solve x² + 6x + 5 = 0:

1. Move the constant: x² + 6x = −5
2. Add (half of 6)² = 9 to both sides: x² + 6x + 9 = 4
3. Write as a square: (x + 3)² = 4
4. Take square roots: x + 3 = ±2
5. So x = −1 or x = −5

The Vertex and the Graph (Parabola)

The graph of y = ax² + bx + c is a parabola. It opens upwards when a > 0 and downwards when a < 0. Its turning point, the vertex, is at:

x = −b ÷ 2a

For y = x² − 5x + 6, the vertex is at x = 5 ÷ 2 = 2.5 and y = 6.25 − 12.5 + 6 = −0.25. The roots 2 and 3 are symmetric around 2.5, which is always true: the vertex lies exactly halfway between the roots.

Sum and Product of Roots

For ax² + bx + c = 0 with roots α and β:

• Sum of roots: α + β = −b ÷ a
• Product of roots: αβ = c ÷ a

For x² − 5x + 6 = 0: sum = 5 and product = 6, matching the roots 2 and 3. These relationships are useful for checking answers and for forming an equation from given roots: x² − (sum)x + (product) = 0.

Real-Life Example: Projectile Motion

A ball is thrown upwards so that its height after t seconds is h = −5t² + 20t metres. When does it land? Set h = 0: −5t² + 20t = 0, so −5t(t − 4) = 0, giving t = 0 (the throw) or t = 4 seconds (landing). The maximum height is at the vertex, t = −20 ÷ (2 × −5) = 2 seconds, where h = −20 + 40 = 20 metres. Quadratics also model areas, profit and revenue, and braking distances.

Which Method Should You Use?

• Factorisation – fastest when the roots are small whole numbers or simple fractions.
• Completing the square – useful for finding the vertex and for deriving results.
• Quadratic formula – always works, including for irrational and complex roots.

In exams, check the discriminant first: if it is a perfect square, try factorising; otherwise go straight to the formula.

Common Mistakes to Avoid

• Forgetting to rearrange the equation to "= 0" before solving.
• Sign errors with a negative b or c – write −b and b² carefully, remembering that (−5)² = 25.
• Dividing only part of the numerator by 2a.
• Dividing both sides by x and losing the root x = 0.
• Not checking answers by substituting them back.

For linear, quadratic and simultaneous equations with full working, try our algebra calculator; for powers, roots and trigonometry use the scientific calculator.

Practice Questions with Answers

1. x² − 7x + 12 = 0 → factors (x − 3)(x − 4), so x = 3 or 4.
2. x² − 9 = 0 → difference of squares (x − 3)(x + 3), so x = ±3.
3. 3x² − 12x = 0 → take out 3x: 3x(x − 4) = 0, so x = 0 or 4.
4. x² + x − 1 = 0 → D = 5, so x = (−1 ± √5) ÷ 2 ≈ 0.618 or −1.618.

Question 4 gives the golden ratio – a reminder that the formula handles irrational roots that factorisation cannot.

Frequently Asked Questions

Note: All examples use real coefficients. The same formula works for complex coefficients, but that is beyond most school syllabuses.