How to Solve Quadratic Equations: Formula & Worked Examples

A quadratic equation has the form ax² + bx + c = 0 (a ≠ 0). There are three ways to solve it: factoring, the quadratic formula x = (−b ± √(b²−4ac)) / 2a, or completing the square. The discriminant D = b² − 4ac determines the nature of the roots: D>0 two distinct roots, D=0 one repeated root, D<0 no real roots.

What Is a Quadratic Equation?

A quadratic equation is an equation with a variable raised to the second power, in the standard form:

where a, b, c are constants and a ≠ 0. The concept dates back to ancient Babylonia around 2000 BC — Babylonian mathematicians were already able to solve problems equivalent to quadratic equations, although they did not use modern notation. The mathematician al-Khwarizmi later formalised the method in the 9th century AD in his work Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala.

The roots of a quadratic equation are the values of x that satisfy the equation. A quadratic equation can have two distinct roots, one repeated root, or no real roots — depending on the value of its discriminant.

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Three Methods for Solving Quadratic Equations

Method 1: Factoring

Find two numbers p and q such that p + q = b and p × q = a×c:

Method 2: The Quadratic Formula

Formula: x = (−b ± √(b²−4ac)) / (2a)

Discriminant :

• D > 0: two distinct roots
• D = 0: one repeated root
• D < 0: no real roots

Method 3: Completing the Square

Rewrite the equation in the form , then take the square root.

💡 Tip for choosing a method: Try factoring first — if you cannot factor within 30 seconds, switch to the quadratic formula. The quadratic formula always works for every quadratic equation and frequently appears in standardised exams.

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How to Solve Quadratic Equations: Worked Examples

Example 1: Factoring

Solve:

Step 1: a=1, b=−5, c=6. Find p and q: p+q = −5, p×q = 6 → p = −2, q = −3 (since −2+(−3)=−5 and −2×(−3)=6)

Step 2: Factor

Step 3: x − 2 = 0 → x = 2, or x − 3 = 0 → x = 3

Answer: x = 2 or x = 3

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Example 2: Quadratic Formula

Solve:

Step 1: a=2, b=−3, c=−2

Step 2: Compute the discriminant

Step 3: Substitute into the quadratic formula

Answer: x = 2 or x = −1/2

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Example 3: Completing the Square

Solve:

Step 1: Move the constant to the right-hand side

Step 2: Add to both sides

Step 3: x + 2 = ±3 → x = 1 or x = −5

Answer: x = 1 or x = −5

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Common Mistakes When Solving Quadratic Equations

Mistake 1: Misreading the sign of a coefficient In , the coefficient b = −5 (negative), not +5. A common error is substituting b = +5 into the quadratic formula, which gives the wrong result. Always check the sign in front of each term before identifying a, b, c.

Mistake 2: Forgetting the ± (plus-minus) in the quadratic formula The quadratic formula produces two roots: x₁ with the + sign and x₂ with the − sign. Dropping one of the signs gives only one root and an incomplete answer. Write both possibilities explicitly: x = (−b + √D) / 2a and x = (−b − √D) / 2a.

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Table of Quadratic Equation Types

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Practice Problems

Problem 1: ⭐ Solve by factoring:

Problem 2: ⭐⭐ Solve using the quadratic formula:

Problem 3: ⭐⭐ Solve by completing the square:

Problem 4: ⭐⭐⭐ A rectangular city park has an area of 60 m². Its length is 4 m greater than its width. Find the dimensions of the park using a quadratic equation!

Problem 5: ⭐⭐⭐ (Exam-style) The quadratic equation has two equal real roots. Find the value of k!

Answer Key:

1. (x−3)(x−4)=0 → x = 3 or x = 4
2. a=1, b=2, c=−8. D=4+32=36. x=(−2±6)/2 → x = 2 or x = −4
3. x²−6x=−5 → (x−3)²=4 → x−3=±2 → x = 5 or x = 1
4. w(w+4)=60 → w²+4w−60=0 → (w+10)(w−6)=0 → w=6, l=10 → length 10 m, width 6 m
5. D = 0 → (k+2)² − 4(2k) = 0 → k² + 4k + 4 − 8k = 0 → k² − 4k + 4 = 0 → (k−2)² = 0 → k = 2

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Summary

1. A quadratic equation has the form ax² + bx + c = 0 (a ≠ 0) and has at most two real roots.
2. Three solution methods: factoring (quick when possible), the quadratic formula (always works), and completing the square (important for conceptual understanding).
3. The discriminant D = b² − 4ac determines the nature of the roots: D>0 two distinct roots, D=0 one repeated root, D<0 no real roots.
4. The quadratic formula is essential for Grade 9 exams and preparation for standardised tests.
5. Quadratic equations are applied in physics (projectile motion), economics (profit optimisation), and engineering (arch design).

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Read also: How to Calculate the Perimeter of a Rectangle and How to Find the Surface Area of a Cylinder.