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CBSE · Class 10 · Mathematics

Quadratic Equations 4.3: Solution by Factorisation

5 concepts · 40 practice questions with answers · Free to read

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Splitting the middle term

From daily life

To open a quadratic into two brackets, break the middle term into two pieces that help the factors appear.

Simple definition

For ax² + bx + c, find two numbers whose product is ac and sum is b. Use them to split the middle term, then take common factors in pairs.

Formula: product = ac; sum = b

x² − 7x + 12−3 and −4product 12, sum −7x² − 3x − 4x + 12(x − 3)(x − 4)

In the figure: x² − 7x + 12: numbers −3 and −4, so x(x − 3) − 4(x − 3) = (x − 3)(x − 4).

Rule 1: Find the pair — Check pairs of factors of ac with the right signs.

  • Easy: x² + 5x + 6: 2 and 3, (x + 2)(x + 3).
  • Medium: x² − x − 12: −4 and 3, (x − 4)(x + 3).
  • Exam-type: x² + 2x − 15: 5 and −3, (x + 5)(x − 3).

Rule 2: Sign rule — If c is positive both numbers have the sign of b; if c is negative they differ.

  • Easy: c = +12, b = −7: both negative.
  • Medium: c = −12, b = −1: one positive, one negative.
  • Exam-type: Check by multiplying the brackets back.
Remember

The two numbers always multiply to ac and add to b: test both quickly before splitting.

Common mistake

Choosing numbers that multiply correctly but do not add to b.

Memory clue

"Product ac, sum b."

Practice questions

Easy
  1. 1To factorise x² + 5x − 36 = 0, we need two numbers whose sum is 5 and product is −36. They are
  2. 2To factorise x² + x − 20 = 0, we need two numbers whose sum is 1 and product is −20. They are
  3. 3To factorise x² + 7x + 10 = 0, we need two numbers whose sum is 7 and product is 10. They are
Medium
  1. 4To factorise x² − 4x − 21 = 0, we need two numbers whose sum is −4 and product is −21. They are
  2. 5To factorise x² + 12x + 32 = 0, we need two numbers whose sum is 12 and product is 32. They are
  3. 6To factorise x² + 6x − 27 = 0, we need two numbers whose sum is 6 and product is −27. They are
Challenging
  1. 7To factorise x² − 6x − 7 = 0, we need two numbers whose sum is −6 and product is −7. They are
  2. 8To factorise x² + 10x + 21 = 0, we need two numbers whose sum is 10 and product is 21. They are
Show answers (try them yourself first)
  1. 1−4 and 9
  2. 2−4 and 5
  3. 35 and 2
  4. 4−7 and 3
  5. 54 and 8
  6. 6−3 and 9
  7. 7−7 and 1
  8. 87 and 3

Roots of x² + bx + c = 0 by factorising

From daily life

If two numbers multiply to zero, at least one of them must be zero. That is why each bracket can be set to zero.

Simple definition

Factorise the quadratic, then use the zero product rule: if (x − p)(x − q) = 0 then x = p or x = q.

Formula: (x − p)(x − q) = 0 ⇒ x = p or q

x² − 5x + 6 = 0(x − 2)(x − 3) = 0x − 2 = 0 or x − 3 = 0x = 2, 3

In the figure: x² − 5x + 6 = (x − 2)(x − 3) = 0, so x = 2 or 3.

Live figure

Roots of x² + bx + c = 0 by factorising

In the app you can move this figure with your finger.

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Rule 1: Solve — Factorise, then set each bracket to zero.

  • Easy: x² + x − 6 = (x + 3)(x − 2) = 0: x = −3, 2.
  • Medium: x² − 9 = (x − 3)(x + 3) = 0: x = ±3.
  • Exam-type: x² − 3x − 10 = (x − 5)(x + 2) = 0: x = 5, −2.

Rule 2: Equal roots — A perfect square gives the same root twice.

  • Easy: x² + 6x + 9 = (x + 3)² = 0: x = −3 twice.
  • Medium: x² − 8x + 16 = (x − 4)² = 0: x = 4 twice.
  • Exam-type: Check: (−3)² + 6(−3) + 9 = 0 ✓.
Remember

The bracket (x + 3) gives the root −3, with the OPPOSITE sign.

Common mistake

Writing the root with the same sign as in the bracket.

Memory clue

"Bracket zero, flip the sign."

Practice questions

Easy
  1. 1Solve x² − 2x − 24 = 0 by factorisation. Find the sum of the roots.
  2. 2Solve x² − 9x + 14 = 0 by factorisation. Find the smaller root.
  3. 3Solve x² − 10x + 16 = 0 by factorisation. Find the product of the roots.
Medium
  1. 4Solve x² + 5x − 36 = 0 by factorisation. Find the larger root.
  2. 5Solve x² − x − 30 = 0 by factorisation. Find the larger root.
  3. 6Solve x² − 13x + 12 = 0 by factorisation. Find the product of the roots.
Challenging
  1. 7Solve x² + 9x − 22 = 0 by factorisation. Find the positive difference of the roots.
  2. 8Solve x² + 4x − 12 = 0 by factorisation. Find the sum of the roots.
Show answers (try them yourself first)
  1. 12
  2. 22
  3. 316
  4. 44
  5. 56
  6. 612
  7. 713
  8. 8−4

When a ≠ 1 and incomplete equations

From daily life

With a leading number like 2x², the pair of numbers must multiply to a × c. Equations missing a term are even quicker.

Simple definition

For a ≠ 1, find numbers with product ac and sum b. For an incomplete equation: ax² + bx = 0 ⇒ x(ax + b) = 0; ax² + c = 0 ⇒ x² = −c/a.

2x² − 7x + 3ac = 6−1 and −62x²−x−6x+3(2x−1)(x−3)

In the figure: 2x² − 7x + 3 = 0: (2x − 1)(x − 3) = 0, so x = 1/2 or 3.

Rule 1: Leading coefficient not 1 — Use ac, then pair-wise common factors.

  • Easy: 2x² − 7x + 3 = 0: x = 1/2, 3.
  • Medium: 6x² − x − 2 = 0: (3x − 2)(2x + 1) = 0, x = 2/3, −1/2.
  • Exam-type: Check: 2(1/2)² − 7(1/2) + 3 = 1/2 − 7/2 + 3 = 0 ✓.

Rule 2: Incomplete equations — No constant: take x common. No middle term: take square roots.

  • Easy: x² − 4x = 0 ⇒ x(x − 4) = 0: x = 0, 4.
  • Medium: 3x² − 12 = 0 ⇒ x² = 4: x = ±2.
  • Exam-type: Never divide by x: that would lose the root 0.
Remember

Check that the two numbers you chose give the right middle coefficient before moving on.

Common mistake

Dividing both sides by x in x² = 4x, which loses the root x = 0.

Memory clue

"Product ac, sum b; take x common if no constant."

Practice questions

Easy
  1. 1The roots of 6x² − 96 = 0 are
  2. 2The roots of x² = 36 are
  3. 3The roots of x² = 81 are
Medium
  1. 4The roots of x² = 121 are
  2. 5The roots of x² = 169 are
  3. 6The roots of x² = 6x are
Challenging
  1. 7The roots of x² = 144 are
  2. 8The roots of x² = 100 are
Show answers (try them yourself first)
  1. 14 and −4
  2. 26 and −6
  3. 39 and −9
  4. 411 and −11
  5. 513 and −13
  6. 60 and 6
  7. 712 and −12
  8. 810 and −10

Word problems by factorisation

From daily life

Form the equation from the story, factorise it, and keep only the root that makes sense: lengths and ages cannot be negative.

Simple definition

After solving, reject roots that do not fit the situation (negative lengths, ages or counts), and answer in words with units.

formequationfactoriserootsrejectand answer

In the figure: Consecutive integers with product 306: x² + x − 306 = 0, (x + 18)(x − 17) = 0, so 17 and 18.

Rule 1: Numbers — Reject a negative root if the numbers must be positive.

  • Easy: x² + x − 306 = 0: x = 17 (reject −18); numbers 17 and 18.
  • Medium: Consecutive even numbers with product 168: (x + 14)(x − 12) = 0: 12 and 14.
  • Exam-type: A number plus its square is 12: (x + 4)(x − 3) = 0: 3 or −4.

Rule 2: Geometry — Lengths must be positive.

  • Easy: Length = breadth + 3, area 40: b² + 3b − 40 = 0, (b + 8)(b − 5) = 0: breadth 5 m, length 8 m.
  • Medium: Number plus reciprocal is 10/3: (3x − 1)(x − 3) = 0: x = 3 or 1/3.
  • Exam-type: Reject b = −8: a length cannot be negative.
Remember

Always test each root in the story, not just in the equation.

Common mistake

Writing both roots as answers, including a negative length.

Memory clue

"Form, factorise, reject the misfit."

Practice questions

Easy
  1. 1The product of two consecutive positive odd integers is 35. Find the smaller one.
  2. 2The product of two consecutive positive integers is 156. Find the larger one.
  3. 3The product of two consecutive positive integers is 600. Find the larger one.
Medium
  1. 4The product of Meera's age 4 years ago and her age 2 years from now is 567. Find her present age (in years).
  2. 5A train covers 75 km at a uniform speed. If its speed were 20 km/h more, it would take 1 hour less. Find its speed (in km/h).
  3. 6The sum of two positive numbers is 34 and their product is 285. Find the smaller number.
Challenging
  1. 7The product of Meera's age 2 years ago and her age 4 years from now is 952. Find her present age (in years).
  2. 8The hypotenuse of a right triangle is 20 cm and one of the other sides is 4 cm longer than the second. Find the shorter of these two sides (in cm).
Show answers (try them yourself first)
  1. 15
  2. 213
  3. 325
  4. 425
  5. 530
  6. 615
  7. 730
  8. 812

Roots, factors and forming the equation

From daily life

Going backwards: if you know the roots, the factors and the equation come right out.

Simple definition

If the roots are p and q, the equation is (x − p)(x − q) = 0, i.e. x² − (p + q)x + pq = 0. For fractions clear the denominators.

Formula: x² − (p + q)x + pq = 0

roots 2, −3(x − 2)(x + 3) = 0x² + x − 6 = 0

In the figure: Roots 2 and −3: (x − 2)(x + 3) = 0, so x² + x − 6 = 0.

Rule 1: Equation from roots — Multiply the two brackets and tidy up.

  • Easy: Roots 1/2 and −1: (2x − 1)(x + 1) = 0 ⇒ 2x² + x − 1 = 0.
  • Medium: Equal roots 4, 4: x² − 8x + 16 = 0.
  • Exam-type: Roots 5 and −5: x² − 25 = 0.

Rule 2: Check with sum and product — The sum of the roots is −b/a and the product is c/a.

  • Easy: 2x² + x − 1 = 0: sum −1/2 ✓, product −1/2 ✓.
  • Medium: x² − 8x + 16 = 0: sum 8 ✓, product 16 ✓.
  • Exam-type: Any non-zero multiple of the equation has the same roots.
Remember

The same roots can come from many equations: multiply by any non-zero constant.

Common mistake

Writing (x + p)(x + q) for roots p and q.

Memory clue

"Roots p, q: factors (x − p)(x − q)."

Practice questions

Easy
  1. 1Which quadratic equation (with leading coefficient 1) has roots −7 and −6?
  2. 2Which quadratic equation (with leading coefficient 1) has roots −5 and −2?
  3. 3Which quadratic equation (with leading coefficient 1) has roots 3 and 9?
Medium
  1. 4Which quadratic equation (with leading coefficient 1) has roots 4 and −1?
  2. 5Which quadratic equation (with leading coefficient 1) has roots 6 and −1?
  3. 6If α and β are the roots of x² − 3x + 2 = 0, find α² + β².
Challenging
  1. 7If α and β are the roots of x² + x − 56 = 0, find (α − β)².
  2. 8If α and β are the roots of x² + 17x + 72 = 0, find α² + β².
Show answers (try them yourself first)
  1. 1x² + 13x + 42 = 0
  2. 2x² + 7x + 10 = 0
  3. 3x² − 12x + 27 = 0
  4. 4x² − 3x − 4 = 0
  5. 5x² − 5x − 6 = 0
  6. 65
  7. 7225
  8. 8145