Quadratic Equations — Class 10 MCQs with Answers

Quadratic Equations — important questions & MCQs with answers (Class 10 Mathematics)

6 solved questions from this chapter's difficulty levels, each with its answer and explanation. Tap an option to check yourself.

0 of 6 answered

1. Q1 of 6Easy

   Which of the following is a quadratic equation?

   Ax² + 2x − 3 = 0B2x − 5 = 0Cx³ − 1 = 0Dx⁴ + 2 = 0
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   Answer: x² + 2x − 3 = 0

   Explanation: A quadratic equation has degree exactly 2, form ax² + bx + c = 0 with a ≠ 0; 2x − 5 = 0 is linear, x³ − 1 = 0 cubic, x⁴ + 2 = 0 quartic — only x² + 2x − 3 = 0 fits the degree-2 pattern.

2. Q2 of 6Easy

   The roots of x² − 5x + 6 = 0 are:

   A−1, −6B−2, −3C1, 6D2, 3
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   Answer: 2, 3

   Explanation: Factorise: x² − 5x + 6 = (x − 2)(x − 3) = 0, giving roots 2 and 3 — check: sum 5 matches −b/a and product 6 matches c/a.

3. Q3 of 6Medium

   The nature of roots of 2x² − 3x + 5 = 0 is:

   ANot realBReal and equalCReal and distinctDReal and both positive
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   Answer: Not real

   Explanation: D = b² − 4ac = (−3)² − 4(2)(5) = 9 − 40 = −31, which is negative, so the roots are not real.

4. Q4 of 6Medium

   Assertion (A): The roots of the equation 9x² – 6x + 1 = 0 are equal. Reason (R): A quadratic equation has two equal real roots if b² – 4ac > 0.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: A is true but R is false

   Explanation: A is true: b² – 4ac = 36 – 36 = 0, so the roots are equal. R is false: equal roots need b² – 4ac = 0, while b² – 4ac > 0 gives two distinct real roots.

5. Q5 of 6Hard

   The quadratic equation 2x² + kx + 2 = 0 has equal roots when k equals:

   A±2B±4C±√2D±2√2
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   Answer: ±4

   Explanation: Equal roots need D = 0: k² − 4(2)(2) = k² − 16 = 0, so k = ±4.

6. Q6 of 6Hard

   Assertion (A): The two consecutive positive integers whose product is 272 are 16 and 17. Reason (R): A quadratic equation ax² + bx + c = 0 has no real roots if b² – 4ac < 0.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: Both A and R are true but R is not the correct explanation of A

   Explanation: A is true: n(n + 1) = 272 gives n² + n – 272 = 0, so (n – 16)(n + 17) = 0 and n = 16. R is also true, but this equation has b² – 4ac = 1089 > 0, so R does not explain A.

Quadratic Equations notes: key concepts (Class 10 Mathematics)

A quadratic equation can have two, one or no real roots, and the discriminant decides which without solving. This chapter solves quadratics by factorisation, by completing the square and by the quadratic formula, and applies them to word problems.

Quadratic equation

An equation of the form ax² + bx + c = 0 with a ≠ 0. The condition a ≠ 0 is what keeps it quadratic rather than linear.

Standard form

Every quadratic must be rearranged to ax² + bx + c = 0 before any method is applied, with all terms on one side.

Root of a quadratic

A value of x satisfying the equation; geometrically it is where the parabola meets the x-axis.

Solving by factorisation

Split the middle term into two parts whose product is ac and whose sum is b, factor in pairs, then set each factor to zero.

Zero-product rule

If a product of two factors is zero then at least one factor is zero — the principle that makes factorisation work.

Completing the square

Rewriting ax² + bx + c as a perfect square plus a constant, which solves the equation and also derives the quadratic formula.

Quadratic formula

A direct expression for both roots in terms of a, b and c, usable on any quadratic including those that will not factorise.

Discriminant

The quantity D = b² − 4ac, whose sign alone reveals the nature of the roots before any solving is done.

Nature of roots

D > 0 gives two distinct real roots, D = 0 gives two equal real roots, and D < 0 gives no real roots at all.

Relation between roots and coefficients

The roots add to −b/a and multiply to c/a, providing a fast check on any answer.

Word problems

Situations involving areas, speeds, ages and numbers that produce a quadratic; reject any root that is physically impossible, such as a negative length.