Quadratic Equations — Class 10 Maths Notes With Examples
Key Points At A Glance
- A quadratic equation has the standard form ax² + bx + c = 0, where a ≠ 0.
- Three main solving methods: factorisation, completing the square, and the quadratic formula.
- The quadratic formula x = (−b ± √(b²−4ac)) / 2a solves every quadratic.
- The discriminant D = b²−4ac tells you how many real roots exist.
- Always check whether the equation is truly quadratic (a must not be 0).
If quadratic equations have ever made your head spin, take a breath — by the end of these notes you'll be able to solve them with confidence. We'll build the idea slowly, with worked examples you can follow on paper.
What Is a Quadratic Equation?
A quadratic equation is any equation that can be written in the standard form:
- ax² + bx + c = 0
Here a, b and c are numbers (called coefficients), and the most important rule is that a ≠ 0. If a were 0, there would be no x² term and it would just be a normal linear equation.
A few examples of quadratic equations:
- x² − 5x + 6 = 0
- 2x² + 3x − 2 = 0
- x² − 9 = 0
The solutions of a quadratic equation are called its roots. A quadratic can have at most two roots.
Method 1 — Solving by Factorisation
Factorisation is usually the quickest method when the numbers are friendly. The idea is to split the middle term.
Example: Solve x² − 5x + 6 = 0
- We need two numbers that multiply to give +6 and add to give −5.
- Those numbers are −2 and −3.
- Rewrite: x² − 2x − 3x + 6 = 0
- Group: x(x − 2) − 3(x − 2) = 0
- Factor: (x − 2)(x − 3) = 0
- So x = 2 or x = 3.
Method 2 — The Quadratic Formula
When an equation does not factorise neatly, this formula always works. For ax² + bx + c = 0:
- x = ( −b ± √(b² − 4ac) ) / 2a
Example: Solve 2x² + 3x − 2 = 0
- Here a = 2, b = 3, c = −2.
- b² − 4ac = 9 − (4 × 2 × −2) = 9 + 16 = 25.
- √25 = 5.
- x = (−3 ± 5) / 4.
- So x = 2/4 = 0.5 or x = −8/4 = −2.
The roots are x = 0.5 and x = −2.
The Discriminant — Reading the Roots Before Solving
The part under the square root, D = b² − 4ac, is called the discriminant. It tells you the nature of the roots before you finish solving:
- If D > 0 → two distinct real roots.
- If D = 0 → two equal real roots (one repeated root).
- If D < 0 → no real roots (the roots are imaginary).
This is a favourite one-mark question in exams, so learn it well.
Method 3 — Completing the Square (The Idea Behind the Formula)
This method turns the equation into a perfect square. It is also where the quadratic formula actually comes from, so understanding it makes the formula feel less like magic.
- Move the constant to the other side.
- Make the coefficient of x² equal to 1.
- Add the square of half the x-coefficient to both sides.
- Write the left side as a perfect square and solve.
Common Mistakes to Avoid
- Forgetting that a ≠ 0 — always confirm the equation is really quadratic.
- Sign errors when splitting the middle term (work slowly).
- Forgetting the ± in the quadratic formula, which loses one root.
- Not simplifying the final answer.
Quick Summary
- Standard form: ax² + bx + c = 0, a ≠ 0.
- Try factorisation first; use the quadratic formula when stuck.
- The discriminant D = b² − 4ac reveals how many real roots exist.
Maths rewards practice more than reading, so solve a few problems by hand right after reading this. When you're ready for a memory-based subject, see our Photosynthesis Class 10 notes, and to revise efficiently use the methods in How to Study Smart for Exams. You can also browse all Mathematics notes any time.
Frequently Asked Questions
It is an equation where the highest power of the variable is 2 — for example x² − 5x + 6 = 0. Its standard form is ax² + bx + c = 0 with a not equal to 0.
Try factorisation first because it is fastest. If the equation does not factorise into nice whole numbers, use the quadratic formula, which always works.
The discriminant D = b² − 4ac tells the nature of the roots. If D is greater than 0 there are two real roots, if D = 0 there is one repeated root, and if D is less than 0 there are no real roots.
Yes. When the discriminant is exactly 0, the two roots become equal, so the equation has one repeated real root.
If a is 0, the x² term disappears and the equation becomes linear (bx + c = 0), not quadratic. The x² term is what makes it a quadratic.
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