Q1. Solve x² − 7x + 12 = 0 by factorisation.
Find two numbers with product 12 and sum −7: −3 and −4
x² − 3x − 4x + 12 = 0 ⇒ x(x − 3) − 4(x − 3) = 0
(x − 3)(x − 4) = 0 ⇒ x = 3 or x = 4
Quadratic Equations — Class 10
Type any quadratic and watch it solved — discriminant, nature of roots and the two solutions, step by step. Then master factorisation for the exam.
Quadratic equationAn equation of the form ax² + bx + c = 0 (a ≠ 0). It has at most two roots.
Quadratic formulax = [−b ± √(b² − 4ac)] / 2a solves any quadratic.
Discriminant (D)D = b² − 4ac decides the nature of roots: D > 0 real & distinct, D = 0 real & equal, D < 0 no real roots.
Sum & product of rootsSum = −b/a, product = c/a — handy for forming or checking equations.
Quadratic equation solver Interactive
Enter a, b and c to solve ax² + bx + c = 0 — with the discriminant, nature of roots and both solutions.
x² + x + = 0
Discriminant D—
Nature of roots—
Root x₁—
Root x₂—
Numericals practice Interactive
Discriminant and nature-of-roots problems with steps.
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Solved examples
Q2. Find the nature of the roots of 2x² − 4x + 3 = 0.
D = b² − 4ac = (−4)² − 4(2)(3) = 16 − 24 = −8
D < 0 ⇒ the equation has no real roots.
Formula sheet
| Quantity | Formula |
|---|---|
| Quadratic formula | x = [−b ± √(b²−4ac)] / 2a |
| Discriminant | D = b² − 4ac |
| Sum of roots | α + β = −b/a |
| Product of roots | α·β = c/a |
Common mistakes & exam wins
- Always write the equation as ax² + bx + c = 0 first, then read off a, b, c.
- Check the discriminant before solving — if D < 0 there are no real roots.
- For factorisation: split the middle term into two numbers whose product is a·c and sum is b.
- A perfect-square trinomial (D = 0) has equal roots x = −b/2a.
Frequently asked questions
What is the quadratic formula?
x = [−b ± √(b² − 4ac)] / 2a, which solves any quadratic equation ax² + bx + c = 0.
What is the discriminant?
D = b² − 4ac. It tells the nature of the roots: two distinct real (D>0), equal real (D=0) or no real roots (D<0).
How do you find the nature of roots without solving?
Compute the discriminant D = b² − 4ac and check its sign.