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CBSE · Class 10 · Mathematics

Quadratic Equations 4.2: Quadratic Equations

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

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Recognising a quadratic equation

From daily life

The area of a square plot of side x is x². Equations in which the unknown is squared appear whenever areas and products are involved.

Simple definition

A quadratic equation in x can be written as ax² + bx + c = 0 with a ≠ 0. The highest power of x is 2.

Formula: ax² + bx + c = 0, a ≠ 0

ax²square term+ bxlinear term+ cconstant= 0

In the figure: x² − 5x + 6 = 0 is quadratic; 2x + 3 = 0 is not (highest power is 1).

Rule 1: Yes or no? — Expand and simplify first; then look at the highest power.

  • Easy: x(x + 2) = 3 ⇒ x² + 2x − 3 = 0: quadratic.
  • Medium: (x + 1)² = 2x ⇒ x² + 1 = 0: quadratic.
  • Exam-type: x³ − x = 0 has degree 3: not quadratic.

Rule 2: Hidden linear cases — If the x² terms cancel, the equation is linear.

  • Easy: x² + 3x = x² − 5: 3x = −5, linear.
  • Medium: (x − 2)² = x² + 4: −4x + 4 = 4 ⇒ x = 0, linear.
  • Exam-type: x² = 9 is quadratic (x = ±3).
Remember

Bring every term to the left side before judging the degree.

Common mistake

Calling x(x + 2) = x² − 3 quadratic although the x² terms cancel.

Memory clue

"Square the unknown, nothing higher."

Practice questions

Easy
  1. 1Is (x − 9)(x − 3) = −9 a quadratic equation in x?
  2. 2Is (x − 3)² = −3 a quadratic equation in x?
  3. 3Is 6x² + 9 = −2x a quadratic equation in x?
Medium
  1. 4Is x(x + 8) = 1 a quadratic equation in x?
  2. 5Is x(x + 4) = 6 a quadratic equation in x?
  3. 6Is x² + 5x = 2 a quadratic equation in x?
Challenging
  1. 7What is the degree of (x + 2)² = x after it is simplified?
  2. 8What is the degree of (x − 7)² = x² + 4 after it is simplified?
Show answers (try them yourself first)
  1. 1Yes
  2. 2Yes
  3. 3Yes
  4. 4Yes
  5. 5Yes
  6. 6Yes
  7. 72
  8. 81

Standard form and coefficients

From daily life

To compare equations fairly, we write them all in the same order: x² term, x term, constant.

Simple definition

Move all terms to one side to get ax² + bx + c = 0. Then a is the coefficient of x², b of x, and c the constant. Keep the signs.

3x² = 5x − 2move terms3x² − 5x + 2 = 0a=3, b=−5, c=2

In the figure: 3x² = 5x − 2 becomes 3x² − 5x + 2 = 0 with a = 3, b = −5, c = 2.

Rule 1: Write in standard form — Move everything to the left; keep the order x², x, constant.

  • Easy: x(x − 4) = 5 ⇒ x² − 4x − 5 = 0.
  • Medium: 2x² = 7 ⇒ 2x² − 7 = 0.
  • Exam-type: (x + 3)² = 16 ⇒ x² + 6x − 7 = 0.

Rule 2: Read a, b, c — Take each sign with its number; a missing term has coefficient 0.

  • Easy: x² − 4x − 5 = 0: a = 1, b = −4, c = −5.
  • Medium: 2x² − 7 = 0: a = 2, b = 0, c = −7.
  • Exam-type: x² + 3x = 0: a = 1, b = 3, c = 0.
Remember

A missing x or constant term means b = 0 or c = 0, not that the equation is not quadratic.

Common mistake

Dropping the sign of b or c after moving terms across the equals sign.

Memory clue

"All on one side, signs stay."

Practice questions

Easy
  1. 1In the quadratic equation −7x² + 2x − 8 = 0, the constant term is
  2. 2In the quadratic equation −5x² + 9x − 1 = 0, the constant term is
  3. 3In the quadratic equation 7x² − 9x − 1 = 0, the coefficient of x² is
Medium
  1. 4In the quadratic equation 2x² − 9x − 5 = 0, the coefficient of x is
  2. 5In the quadratic equation −2x² − 9x − 3 = 0, the constant term is
  3. 6In the quadratic equation −6x² − 6x − 4 = 0, the coefficient of x is
Challenging
  1. 7In the quadratic equation x² − 7x + 3 = 0, the coefficient of x² is
  2. 8In the quadratic equation −7x² + 9x − 3 = 0, the constant term is
Show answers (try them yourself first)
  1. 1−8
  2. 2−1
  3. 37
  4. 4−9
  5. 5−3
  6. 6−6
  7. 71
  8. 8−3

Forming equations from situations

From daily life

Plots, ages and numbers: whenever two linked quantities multiply to a given result, a quadratic equation appears.

Simple definition

Let the unknown be x, write the other quantity in terms of x, form the product or relation, and bring it to the form ax² + bx + c = 0.

let xthe unknownexpressother in xrelateproductstandardax²+bx+c=0

In the figure: Product of two consecutive integers is 306: x(x + 1) = 306, so x² + x − 306 = 0.

Rule 1: Numbers — Consecutive numbers: x and x + 1 (or x + 2 for even or odd).

  • Easy: Product of consecutive even numbers is 168: x² + 2x − 168 = 0.
  • Medium: A number plus its square is 12: x² + x − 12 = 0.
  • Exam-type: A number plus its reciprocal is 10/3: 3x² − 10x + 3 = 0.

Rule 2: Areas — Area = length × breadth.

  • Easy: Length is 3 m more than breadth, area 40 m²: x(x + 3) = 40, x² + 3x − 40 = 0.
  • Medium: Square of side x has area 49: x² − 49 = 0.
  • Exam-type: A path of width x around a 10 m by 10 m garden: (10 + 2x)² = 196.
Remember

Solving the equation later tells which root fits the story, but first just form it correctly.

Common mistake

Writing consecutive numbers as x and x + 2 for ordinary integers.

Memory clue

"Name x, express the other, multiply."

Practice questions

Easy
  1. 1The length of a rectangle is 5 m more than its breadth and its area is 374 m². Taking the breadth as x metres, the equation is
  2. 2The length of a rectangle is 3 m more than its breadth and its area is 10 m². Taking the breadth as x metres, the equation is
  3. 3The length of a rectangle is 1 m more than its breadth and its area is 342 m². Taking the breadth as x metres, the equation is
Medium
  1. 4The product of two consecutive positive integers is 272. If the smaller is x, the equation is
  2. 5The length of a rectangle is 3 m more than its breadth and its area is 378 m². Taking the breadth as x metres, the equation is
  3. 6The product of two consecutive positive integers is 30. If the smaller is x, the equation is
Challenging
  1. 7The length of a rectangle is 4 m more than its breadth and its area is 96 m². Taking the breadth as x metres, the equation is
  2. 8The length of a rectangle is 3 m more than its breadth and its area is 418 m². Taking the breadth as x metres, the equation is
Show answers (try them yourself first)
  1. 1x² + 5x − 374 = 0
  2. 2x² + 3x − 10 = 0
  3. 3x² + x − 342 = 0
  4. 4x² + x − 272 = 0
  5. 5x² + 3x − 378 = 0
  6. 6x² + x − 30 = 0
  7. 7x² + 4x − 96 = 0
  8. 8x² + 3x − 418 = 0

Roots and checking a number

From daily life

A key fits a lock if it turns it. A number fits an equation if putting it in makes the two sides equal.

Simple definition

The roots of an equation are the values of x for which it is true. A quadratic has at most two roots. To check a number, substitute it and see whether the left side becomes 0.

x = 2put in x²−5x+64−10+6= 0 ✓ root

In the figure: For x² − 5x + 6 = 0: x = 2 gives 4 − 10 + 6 = 0 ✓, x = 3 gives 0 ✓, but x = 1 gives 2 ✗.

Live figure

Roots and checking a number

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Open in the app

Rule 1: Is it a root? — Substitute and compute.

  • Easy: x² − 1 = 0, x = −1: 1 − 1 = 0 ✓ root.
  • Medium: 2x² − 3x − 2 = 0, x = 2: 8 − 6 − 2 = 0 ✓ root.
  • Exam-type: x² − 4x + 3 = 0, x = 2: 4 − 8 + 3 = −1 ✗ not a root.

Rule 2: No real roots — Some equations are never true for any real x.

  • Easy: x² + 4 = 0: x² cannot be −4, no real root.
  • Medium: x² = 9 has roots 3 and −3.
  • Exam-type: x² = 0 has the single root 0 (counted twice).
Remember

A root of a quadratic is also called a zero of the polynomial ax² + bx + c.

Common mistake

Checking a root in the original wording of a problem but not in the equation.

Memory clue

"Put it in; zero out means root."

Practice questions

Easy
  1. 1Is x = 8 a root of x² − 15x + 56 = 0?
  2. 2Is x = 7 a root of x² + 2x − 63 = 0?
  3. 3How many real roots does x² − 9 = 0 have?
Medium
  1. 4Is x = −3 a root of x² + 5x + 6 = 0?
  2. 5Is x = 5 a root of x² − 7x + 6 = 0?
  3. 6How many real roots does x² − 12x + 36 = 0 have?
Challenging
  1. 7How many real roots does x² + 4 = 0 have?
  2. 8How many real roots does x² − 36 = 0 have?
Show answers (try them yourself first)
  1. 1Yes
  2. 2Yes
  3. 32
  4. 4Yes
  5. 5No
  6. 61
  7. 70
  8. 82

Unknown constants and the condition a ≠ 0

From daily life

If a root is given, you can find a missing number in the equation. And the x² term must not vanish, or it is not quadratic.

Simple definition

A given root can be substituted to find an unknown constant. For ax² + bx + c = 0 to be quadratic, the coefficient of x² must not be 0.

x = 1 is a rootof x²+kx−6=01 + k − 6 = 0k = 5

In the figure: If x = 1 is a root of x² + kx − 6 = 0, then 1 + k − 6 = 0, so k = 5.

Rule 1: Find the constant — Put the root in, then solve the small equation.

  • Easy: x = 2 is a root of 3x² − x + k = 0: 12 − 2 + k = 0, k = −10.
  • Medium: x = −1 is a root of x² + 3x + k = 0: 1 − 3 + k = 0, k = 2.
  • Exam-type: x = 3 is a root of x² − kx + 6 = 0: 9 − 3k + 6 = 0, k = 5.

Rule 2: The condition a ≠ 0 — Find the k that kills the x² term and exclude it.

  • Easy: (k − 2)x² + 3x + 1 = 0 is quadratic only if k ≠ 2.
  • Medium: kx² − 5x + 2 = 0 needs k ≠ 0.
  • Exam-type: For k = 2 the first becomes 3x + 1 = 0, a linear equation.
Remember

Always test the final value of k in the original equation.

Common mistake

Forgetting to exclude the k that makes a = 0.

Memory clue

"Root in, solve for k; keep a away from zero."

Practice questions

Easy
  1. 1For what value of k does (k + 4)x² + 5x + 3 = 0 fail to be a quadratic equation?
  2. 2For what value of k does (k + 9)x² + 4x − 6 = 0 fail to be a quadratic equation?
  3. 3For what value of k does (k + 1)x² − 4x − 4 = 0 fail to be a quadratic equation?
Medium
  1. 4If x = −1 is a root of kx² − 9x + 5 = 0, find k.
  2. 5If x = 5 is a root of x² + 2x + k = 0, find k.
  3. 6If x = 9 and x = 8 are the roots of x² + px + q = 0, find the value of q.
Challenging
  1. 7If x = −8 and x = −3 are the roots of x² + px + q = 0, find the value of p.
  2. 8One root of x² − 7x + k = 0 is 4. Find the other root.
Show answers (try them yourself first)
  1. 1−4
  2. 2−9
  3. 3−1
  4. 4−14
  5. 5−35
  6. 672
  7. 711
  8. 83