Quadratic Equations — Roots, Formula and Discriminant
Learn what quadratic equations are and how to solve them by factorisation, completing the square and the quadratic formula, plus using the discriminant to judge the nature of roots — with a worked example and a self-check quiz.
By the PadhoDost Team · 📖 7 min read · Updated 4 August 2026
Part of Class 10 (CBSE) prep🧠 Think of a Cricket Ball's Path
When a batsman lifts the ball for a six, its path through the air rises and then falls in a smooth curve called a parabola. The height of that ball at any moment can be written as a quadratic equation. Any time a quantity depends on the square of something — the area of a plot, the height of a thrown object — you are looking at a quadratic. Solving the equation tells you exactly where that curve meets the ground.
What Is a Quadratic Equation?
A quadratic equation is a polynomial equation in one variable where the highest power of the variable is 2, so it always has an x^2 term. The values of x that make the equation true are called its roots or solutions. A quadratic equation has at most two roots.
Three Ways to Find the Roots
The main methods
- 1Factorisation: split the middle term so the equation becomes a product of two brackets, then set each bracket equal to zero.
- 2Completing the square: rearrange the equation into a perfect square plus a constant, then take square roots on both sides.
- 3Quadratic formula: put the values of a, b and c straight into a formula — this always works, even when factorisation is hard.
| Discriminant D = b^2 - 4ac | Nature of the roots |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | Two equal (coincident) real roots |
| D < 0 | No real roots |
📝 Worked example: a rectangular plot
A rectangular plot's length is 3 m more than its width, and its area is 54 m^2. Find the width.
Let the width be x metres, so the length is (x + 3) metres.
Area = length x width, so x(x + 3) = 54
Bring to standard form: x^2 + 3x - 54 = 0
Split the middle term (two numbers with product -54 and sum +3): +9 and -6
Factorise: (x + 9)(x - 6) = 0
So x = -9 or x = 6. A width cannot be negative, so x = 6.
Width = 6 m and length = 9 m. Check: 6 x 9 = 54 m^2. Correct.
Remember these
- ✓Standard form is ax^2 + bx + c = 0 with a not equal to 0.
- ✓A quadratic equation has at most two real roots.
- ✓The quadratic formula x = (-b plus or minus square root of (b^2 - 4ac))/2a works for every quadratic.
- ✓Discriminant D = b^2 - 4ac decides the nature of roots: D>0 two roots, D=0 equal roots, D<0 no real roots.
- ✓Sum of roots = -b/a and product of roots = c/a.
⚡ Quick check
For the equation 2x^2 - 4x + 5 = 0, what is the nature of its roots?
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