If there is one formula that every Class 10 student needs to master, it is the quadratic formula. Whether you are preparing for CBSE Board exams, ICSE, or any state board, this formula will appear — and understanding it deeply (not just memorising it) is what separates students who score 90+ from those who do not.
What is the Quadratic Formula?
A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a ≠ 0. The quadratic formula gives us the values of x (called the roots) that satisfy this equation:
x = (−b ± √(b²−4ac)) / 2a
It works for every quadratic equation — even ones that cannot be factored easily — making it the most powerful tool in your algebra toolkit.
Understanding Each Part of the Formula
| Part | Variable | What it represents |
|---|---|---|
| −b | b | Coefficient of x (with sign flipped) |
| ± | — | Gives two possible roots (one with +, one with −) |
| √(b²−4ac) | Discriminant | Tells us how many real solutions exist |
| 2a | a | Coefficient of x² (doubles as the denominator) |
What is the Discriminant (D)?
The expression D = b² − 4ac is called the discriminant because it discriminates between different types of roots:
- D > 0: Two distinct real roots
- D = 0: Two equal real roots (the graph just touches the x-axis)
- D < 0: No real roots (complex/imaginary roots)
How to Derive the Quadratic Formula
The formula comes from completing the square on the general form ax² + bx + c = 0:
- Start with: ax² + bx + c = 0
- Divide everything by a: x² + (b/a)x + (c/a) = 0
- Move c/a to the right: x² + (b/a)x = −c/a
- Complete the square (add b²/4a² to both sides): (x + b/2a)² = b²/4a² − c/a
- Simplify the right side: (x + b/2a)² = (b² − 4ac)/4a²
- Take the square root: x + b/2a = ±√(b² − 4ac)/2a
- Subtract b/2a: x = (−b ± √(b² − 4ac))/2a ✓
Step-by-Step Solved Examples
Example 1 (Easy): Solve x² − 5x + 6 = 0
Here a = 1, b = −5, c = 6
D = (−5)² − 4(1)(6) = 25 − 24 = 1
x = (5 ± √1)/2 = (5 ± 1)/2
x = 3 or x = 2 ✓
Example 2 (Medium): A rectangular plot has area 48 m² and its length is 2 m more than twice its width. Find the dimensions.
Let width = x metres, so length = 2x + 2
x(2x + 2) = 48 → 2x² + 2x − 48 = 0 → x² + x − 24 = 0
D = 1 + 96 = 97
x = (−1 + √97)/2 ≈ (−1 + 9.85)/2 ≈ 4.42 m
Width ≈ 4.42 m, Length ≈ 10.85 m ✓
Example 3 (Application): A shopkeeper in Chennai buys a product for ₹x and sells it at ₹(x + 20). If her profit percentage equals her cost price in rupees, find the cost price.
Profit% = Profit/CP × 100, so x = 20/x × 100 → x² = 2000 → x = √2000 ≈ ₹44.72
Verify: CP = ₹44.72, Profit% ≈ 44.72% ✓
Common Mistakes Students Make
- Forgetting the ± sign: Always write both + and − to get both roots, even if one turns out negative.
- Wrong sign for b: The formula has −b, not b. If b is already negative (e.g. b = −5), then −b = +5.
- Dividing only −b by 2a: The entire numerator (−b ± √D) must be divided by 2a, not just −b.
- Square-rooting the discriminant incorrectly: √(b² − 4ac) is NOT the same as b − 2√(ac). Do not split the square root.
- Not checking D first: If D < 0, stop — there are no real roots. Do not proceed to try to compute them.
Real-World Applications
Quadratic equations appear everywhere in the real world:
- Cricket: The path of a ball hit by a batsman follows a parabola — modelled by a quadratic equation.
- Architecture: The shape of the Gateway of India arch is approximately parabolic.
- Finance: Calculating compound interest periods often leads to quadratic equations.
- Farming: Optimising the area of a field for a given fencing budget involves quadratics.
Tips to Remember the Formula
The "minus b plus or minus" rhyme: Many students remember it as a song — "Negative b, plus or minus the square root, of b squared minus four a c, all over two a."
The discriminant trick: Think of D as the "decider" — D decides whether your equation has solutions and how many.
Practice Problems
- (Easy) Solve 2x² − 7x + 3 = 0
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x = 3 or x = 0.5 - (Medium) The sum of a number and its reciprocal is 2.5. Find the number.
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x² − 2.5x + 1 = 0 → x = 2 or x = 0.5 - (Hard) A train travels 360 km. If its speed were 5 km/h more, the journey would take 1 hour less. Find the original speed.
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Speed = 40 km/h (using the quadratic 360/v − 360/(v+5) = 1)
Frequently Asked Questions
Can I use the quadratic formula for any quadratic equation?
Yes — the quadratic formula works for every quadratic equation ax² + bx + c = 0 where a ≠ 0. It is especially useful when an equation cannot be factored easily.
What if the discriminant is negative?
If D = b² − 4ac is negative, the equation has no real roots. In Class 10, this usually means "no solution" in the real number system. In higher classes, you will learn about complex numbers where solutions still exist.
Is the quadratic formula in the CBSE Class 10 syllabus?
Yes, it is a core topic in Chapter 4 (Quadratic Equations) of the CBSE Class 10 Mathematics syllabus and appears regularly in board exams, usually worth 3-4 marks.
Practice the quadratic formula interactively on MathVis — enter any values of a, b, c and see the roots calculated step by step, with a visual graph showing exactly where those roots sit on the parabola.