How to Solve Quadratic Equations (3 Methods)
A quadratic equation is any equation that can be written as ax² + bx + c = 0, where a is not zero. Learning how to solve quadratic equations is essential for CBSE and ICSE board exams and for competitive tests like JEE. In this guide we cover three trusted methods: factoring, completing the square, and the quadratic formula, plus the discriminant that tells you what kind of roots to expect.
Method 1: Factoring
Factoring works best when the equation splits neatly. To solve x² + 5x + 6 = 0, look for two numbers that multiply to give 6 (the constant) and add to give 5 (the middle coefficient). Those numbers are 2 and 3.
- Step 1: Write x² + 5x + 6 = (x + 2)(x + 3) = 0.
- Step 2: Set each factor to zero: x + 2 = 0 or x + 3 = 0.
- Step 3: Solve to get x = −2 or x = −3.
Check: (−2)² + 5(−2) + 6 = 4 − 10 + 6 = 0. Correct.
Method 2: Completing the square
This method always works, even when factoring is hard. Solve x² + 6x + 5 = 0. Move the constant: x² + 6x = −5. Take half of the coefficient of x, which is 6 ÷ 2 = 3, then square it to get 9. Add 9 to both sides:
- Step 1: x² + 6x + 9 = −5 + 9, so x² + 6x + 9 = 4.
- Step 2: The left side is a perfect square: (x + 3)² = 4.
- Step 3: Take the square root: x + 3 = ±2.
- Step 4: So x = −3 + 2 = −1 or x = −3 − 2 = −5.
Method 3: The quadratic formula
For any equation ax² + bx + c = 0, the roots are given by x = [−b ± √(b² − 4ac)] ÷ 2a. This formula never fails. Solve 2x² + 3x − 2 = 0, where a = 2, b = 3, c = −2.
- Step 1: Find b² − 4ac = 3² − 4(2)(−2) = 9 + 16 = 25.
- Step 2: √25 = 5, so x = (−3 ± 5) ÷ 4.
- Step 3: x = (−3 + 5) ÷ 4 = 2 ÷ 4 = 1/2, or x = (−3 − 5) ÷ 4 = −8 ÷ 4 = −2.
The discriminant
The expression b² − 4ac inside the square root is called the discriminant, written as D. It reveals the nature of the roots without fully solving:
- D > 0: two distinct real roots.
- D = 0: two equal real roots (one repeated root).
- D < 0: no real roots (the roots are complex).
For example, in x² − 4x + 4 = 0, D = (−4)² − 4(1)(4) = 16 − 16 = 0, so there is one repeated root, x = 2.
Quadratics appear throughout Class 10 Maths and again in Class 11 Maths. If you plan to attempt engineering entrance tests, strong quadratic skills are vital for JEE Maths. Before this topic, make sure your algebra basics are solid.
Frequently asked questions
Which method should I use to solve a quadratic equation?
Try factoring first because it is fastest. If the equation does not factor easily, use the quadratic formula, which always works. Completing the square is useful for deriving the formula and for certain calculus topics.
What does the discriminant tell us?
The discriminant D = b² − 4ac tells you the nature of the roots: positive means two different real roots, zero means one repeated real root, and negative means no real roots.
Can every quadratic equation be solved by factoring?
No. Many quadratics have irrational or complex roots that do not factor with whole numbers. In those cases use the quadratic formula or completing the square instead.