Quadratic Equations Notes Class 10 Maths
Quadratic Equations
Why this chapter matters
Water flows, balls fly, and area covers a garden — all captured by one equation.
By the end you will turn real-life problems into ax² + bx + c = 0 and solve them.
Chapter Learning Map
Follow the spine top to bottom. 3 groups, all equally important.
1 · Standard Form of a Quadratic Equation
A quadratic equation in x is an equation that can be written as ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
Formula: ax² + bx + c = 0. Rule: highest power is 2, and a cannot be zero.
2x² + 3x − 5 = 0 is quadratic: here a = 2, b = 3, c = −5.
x² = 7 becomes x² − 7 = 0: now a = 1, b = 0, c = −7. It is still quadratic.
Power two is the boss. If x power two appears, watch for a quadratic.
If a = 0, the x² term disappears and the equation becomes linear, not quadratic.
The path of a football kick is a parabola — the graph of a quadratic equation.
2 · Solving by Factorisation
Factor the quadratic into two brackets, set each bracket equal to zero, and solve for x.
Rule: if p × q = 0, then p = 0 or q = 0.
x² − 5x + 6 = (x − 2)(x − 3). So x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.
Factor first, bracket then zero, two answers are the hero.
Never divide both sides by x. It silently destroys the root x = 0.
Checking roots needs 5 seconds: 2² − 5(2) + 6 = 4 − 10 + 6 = 0. Always verify.
3 · The Quadratic Formula
When factorisation is hard, one formula solves every quadratic with real roots.
Formula: x = −b ± √(b² − 4ac)2a . Use it after writing ax² + bx + c = 0.
2x² − 7x + 3 = 0: a = 2, b = −7, c = 3. Then x = 7 ± √(49 − 24)4 = 7 ± 54 = 3 or 12.
Minus b, plus minus root of b² − 4ac, all over 2a — say it once a day.
Write b with its sign. For b = −7, the formula starts with 7, not −7.
The formula works for every quadratic; factorisation is just the fast path when it exists.
4 · Discriminant and Nature of Roots
The discriminant D = b² − 4ac decides how many real roots exist.
x² − 4x + 4 = 0: D = 16 − 16 = 0, so both roots are equal: x = 2, 2.
| Discriminant | Nature of roots | Example |
|---|---|---|
| D > 0 | Two distinct real roots | x² − 5x + 6 = 0 (D = 1) |
| D = 0 | Two equal real roots | x² − 4x + 4 = 0 (D = 0) |
| D < 0 | No real roots | x² + 1 = 0 (D = −4) |
Positive splits, zero repeats, negative retreats — nature of roots in one line.
D < 0 still has complex roots. For Class 10 with real roots, just say “no real roots”.
The curve touches the x-axis once for D = 0, and never touches it for D < 0.
Solved NCERT Highlights
Show solution
Expand left: (x + 1)² = x² + 2x + 1
Expand right: 2(x − 3) = 2x − 6
Bring together: x² + 2x + 1 = 2x − 6
Cancel 2x: x² + 1 = −6, so x² + 7 = 0
Form: 1x² + 0x + 7 = 0 with a = 1 ≠ 0. Yes, it is quadratic.
Show solution
Let base = x cm: then altitude = (x − 7) cm.
Pythagoras: x² + (x − 7)² = 13²
Expand: x² + x² − 14x + 49 = 169
Simplify: 2x² − 14x − 120 = 0, so x² − 7x − 60 = 0
Factor: (x − 12)(x + 5) = 0, so x = 12 or x = −5
Answer: base = 12 cm, altitude = 5 cm.
Check: 12² + 5² = 144 + 25 = 169 = 13² ✔
Topper Secrets & Extra Examples
Sum of roots = −ba and product of roots = ca . Check answers in one line.
For x² − 5x + 6 = 0, factor check: sum 5, product 6, roots 2 and 3. Signs are the same when c is positive.
If c = 0, one root is always x = 0. Factor x out first and never lose it.
In word problems use the meaningful positive root; reject the negative answer with a reason.
x² − 5x + 6 = 0 gives two distinct roots (2 and 3), while x² − 4x + 4 = 0 gives one repeated root (2). The discriminant tells them apart: 1 vs 0.
The sum of the squares of two consecutive positive integers is 365. Let the numbers be n and n + 1. Then n² + (n + 1)² = 365, so 2n² + 2n − 364 = 0, giving n² + n − 182 = 0. Factor (n + 14)(n − 13) = 0, so n = 13. The numbers are 13 and 14. Check: 169 + 196 = 365 ✔
When c is negative the factors have opposite signs. x² − 1 = 0 → (x − 1)(x + 1), roots 1 and −1.
Symbol / Diagram Board
One shape only: the parabola y = ax² opens up when a > 0 and down when a < 0.
The vertex sits on the axis of symmetry. Roots are where the curve meets the x-axis.
⚡ Formula Vault
Revise these before every test. Each carries marks.
Important Question Bank
Questions in sequence. Tap “Show answer”.
A · Multiple Choice (1 mark each)
(a) 1 and −3 (b) −1 and 3 (c) 1 and 3 (d) −1 and −3
Show answer
Answer: (a). x² + 2x − 3 = (x + 3)(x − 1), so x = −3 or x = 1.
(a) 0 (b) 16 (c) 4 (d) −4
Show answer
Answer: (a). D = (−4)² − 4 × 2 × 2 = 16 − 16 = 0.
(a) 2 (b) −2 (c) 43 (d) −43
Show answer
Answer: (a). Sum = −b/a = −(−6)/3 = 6/3 = 2.
(a) 6 (b) 5 (c) −6 (d) 10
Show answer
Answer: (a). Put x = 2: 4 − 10 + k = 0, so k = 6.
(a) no real roots (b) two distinct roots (c) two equal real roots (d) one irrational root
Show answer
Answer: (c). D = 16 − 16 = 0, so the two roots are equal.
B · Assertion–Reason (1 mark each)
(a) Both true, R explains A. (b) Both true, R doesn’t explain A. (c) A true, R false. (d) A false, R true.
Assertion: The equation x² + 1 = 0 has no real roots.
Reason: Its discriminant D = −4 is less than zero.
Show answer
Answer: (a). D = 0 − 4(1)(1) = −4 < 0, and D < 0 means no real roots. The reason explains the assertion.
Assertion: The product of the roots of ax² + bx + c = 0 is c/a.
Reason: The sum of the roots is −b/a.
Show answer
Answer: (b). Both statements are true, but the reason describes the sum, not the product, so it does not explain the assertion.
Assertion: x(x − 1) = 0 is a quadratic equation with roots 0 and 1.
Reason: Its highest power of x is 2.
Show answer
Answer: (a). x(x − 1) = x² − x, a quadratic, and x = 0 or x = 1 satisfy it. The degree explains it.
C · Case-Based (4 marks)
A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, the same journey would have taken 1 hour less.
(i) Taking the usual speed as v km/h, write the time at the usual speed. (ii) Write the time at the faster speed. (iii) Form the quadratic equation in v. (iv) Find the usual speed of the train.
Show answer
(i) Time = 360/v hours.
(ii) Time = 360/(v + 5) hours.
(iii) 360/v − 360/(v + 5) = 1, so v² + 5v − 1800 = 0.
(iv) (v − 40)(v + 45) = 0, so v = 40 km/h.
Check: 360/40 = 9 h and 360/45 = 8 h, one hour less ✔
D · Short & Long Answer
Show answer
Let one number be n: the other is 27 − n.
Equation: n(27 − n) = 182, so n² − 27n + 182 = 0.
Factor: (n − 13)(n − 14) = 0, so n = 13 or n = 14.
Answer: The numbers are 13 and 14.
Show answer
Let breadth = x m: then length = (x + 4) m.
Equation: x(x + 4) = 320, so x² + 4x − 320 = 0.
Factor: (x + 20)(x − 16) = 0, so x = −20 or x = 16.
Reject x = −20 because breadth cannot be negative.
Answer: breadth = 16 m, length = 20 m.
Check: 16 × 20 = 320 m² ✔
E · Previous-Year Style
Show answer
Factor: 6x² − x − 1 = (3x + 1)(2x − 1).
Solve: 3x + 1 = 0 gives x = −1/3; 2x − 1 = 0 gives x = 1/2.
Common Mistakes Table
| Mistake students make | Correct idea |
|---|---|
| Setting a = 0 “because the term is small” | a ≠ 0 is compulsory; otherwise the equation is linear. |
| Dividing both sides by x | This deletes the root x = 0. Factor x out instead. |
| Treating −b as +b in the formula | Carry b’s sign into the formula; −(−7) becomes +7. |
| Accepting a negative length or speed | Reject it and justify: lengths and speeds are positive. |
Active Recall & Revision
Cover answers. Say aloud. Then check.
Q: What does D = b² − 4ac tell you?
D > 0 gives two distinct real roots, D = 0 gives equal real roots, D < 0 gives no real roots.
Q: Write the sum and product of roots of 2x² + 8x − 3 = 0.
Sum = −b/a = −8/2 = −4; product = c/a = −3/2.
Q: Why do quadratic equations usually give two answers?
Because x² has degree two; the graph is a parabola that can meet the x-axis at up to two points.
Revise after 2 days, then 3 days, then 5 days, then 7 days. This locks the memory.