Quadratic Equations: Formula, Methods & Solved Examples
Quadratic equations form one of the highest-scoring chapters in Class 10 mathematics, and the skills you build here carry straight into Class 11 and 12. Once you can recognise a quadratic and pick the right method to solve it, a whole set of board questions become almost mechanical. This guide takes you through the standard form, every solving method your syllabus expects, the discriminant, and fully worked examples in the style of CBSE and state-board papers.
What is a quadratic equation?
A quadratic equation is a polynomial equation in which the highest power of the variable is two. Its general or standard form is written as:
ax² + bx + c = 0 (where a ≠ 0)
Here a, b and c are real numbers and x is the unknown. The condition a ≠ 0 is essential — if a were zero, the x² term would vanish and you would be left with a simple linear equation. The values of x that satisfy the equation are called its roots or solutions, and a quadratic always has at most two of them.
Method 1: Solving by factorisation
This is usually the quickest method when the equation factorises neatly. The idea is to split the middle term so the expression becomes a product of two brackets, then use the fact that if a product is zero, at least one factor must be zero.
Example 1: Solve x² − 7x + 12 = 0
You need two numbers that multiply to give +12 (the value of a × c) and add to give −7 (the value of b). Those numbers are −3 and −4. So we write:
x² − 3x − 4x + 12 = 0, which groups into x(x − 3) − 4(x − 3) = 0, giving (x − 3)(x − 4) = 0. Therefore x = 3 or x = 4.
Method 2: Completing the square
When factorisation is awkward, you can force the left side into a perfect square. Take half the coefficient of x, square it, and add it to both sides.
Example 2: Solve x² + 6x − 7 = 0
Move the constant across: x² + 6x = 7. Half of 6 is 3, and 3² = 9, so add 9 to both sides: x² + 6x + 9 = 16. The left side is now a perfect square: (x + 3)² = 16. Taking the square root, x + 3 = ±4, so x = 1 or x = −7. The roots are 1 and −7.
Method 3: The quadratic formula
The quadratic formula works for every quadratic, even those that do not factorise. It is derived by completing the square on the general form ax² + bx + c = 0:
x = (−b ± √(b² − 4ac)) ÷ 2a
Example 3: Solve 2x² − 4x − 3 = 0
Here a = 2, b = −4, c = −3. First compute b² − 4ac = (−4)² − 4(2)(−3) = 16 + 24 = 40. Then x = (4 ± √40) ÷ 4 = (4 ± 2√10) ÷ 4 = (2 ± √10) ÷ 2. So the two roots are (2 + √10) ÷ 2 and (2 − √10) ÷ 2.
The discriminant and the nature of roots
The expression under the square root, b² − 4ac, is called the discriminant, written as D. Without solving the whole equation, D tells you what kind of roots to expect:
| Value of D = b² − 4ac | Nature of roots |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | Two equal real roots (repeated root) |
| D < 0 | No real roots (roots are imaginary) |
For example, in x² + x + 1 = 0 the discriminant is 1 − 4 = −3, which is negative, so this equation has no real roots — a fact you can state instantly without any calculation.
Relationship between roots and coefficients
If a quadratic ax² + bx + c = 0 has roots α and β, then their sum and product are linked to the coefficients:
- Sum of roots: α + β = −b ÷ a
- Product of roots: α × β = c ÷ a
This lets you build an equation from given roots, or check your answers quickly. If you know the roots are 3 and 4, the equation is x² − (3 + 4)x + (3 × 4) = x² − 7x + 12 = 0.
Choosing the right method
- Try factorisation first — it is fastest when the numbers split cleanly.
- Use completing the square when you are asked for it or when deriving the formula.
- Fall back on the quadratic formula whenever factors are not obvious; it never fails.
Want to test yourself? Try the LearnIQ practice quiz for more mathematics questions, or browse all study guides.