ax²+bx+c

Chapter 4 · Class X Mathematics · MCQ Practice

MCQ Practice Arena

Quadratic Equations

The Discriminant is Your Compass — Navigate Every Quadratic with Confidence

📋 50 MCQs ⭐ 30 PYQs ⏱ 70 sec/Q

🚀 Start Practising 📖 Revise Notes First

MCQ Bank Snapshot

50Total MCQs

18Easy

22Medium

10Hard

30PYQs

70 secAvg Time/Q

7Topics

Easy 36% Medium 44% Hard 20%

Why Practise These MCQs?

CBSE Class XNTSEState BoardsOlympiad

Quadratic Equations MCQs span factorisation speed, discriminant analysis, and word problem modelling. CBSE Boards guarantee 2 marks from "nature of roots" (discriminant MCQ) and 4 marks from a word problem. NTSE uses quadratics in number and geometry contexts. Completing the square is tested in both derivation and application form.

Topic-wise MCQ Breakdown

Recognising Quadratic Form4 Q

Factorisation Method10 Q

Completing the Square6 Q

Quadratic Formula8 Q

Discriminant & Nature of Roots12 Q

Sum & Product of Roots5 Q

Word Problems5 Q

Must-Know Formulae Before You Start

Recall these cold before attempting MCQs — they appear in >70% of questions.

$x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$

$D = b^2 - 4ac$

$D > 0:\ \text{two distinct real roots}$

$D = 0:\ \text{two equal roots, } x = -b/2a$

$D < 0:\ \text{no real roots}$

$\alpha+\beta = -b/a,\ \alpha\beta = c/a$

MCQ Solving Strategy

For factorisation MCQs, find two numbers that multiply to ac and add to b, then split the middle term. For "nature of roots" MCQs, compute D = b²−4ac and check sign — this is 2 free marks. For word problems, define x as the unknown, form the quadratic, solve, and verify the answer fits the context (no negative lengths, ages, etc.). Completing the square MCQs: always move the constant to the right first.

⚠ Common Traps & Errors

⚠D = 0 gives ONE repeated root, not "no roots"
⚠D < 0 means no REAL roots — complex roots exist, but not tested here
⚠"Find k for equal roots" means set D = 0, not D > 0
⚠Quadratic formula: denominator is 2a, not a
⚠Word problem solutions: reject negative values for lengths/ages

Difficulty Ladder

Work through each rung in order — do not jump to Hard before mastering Easy.

① Easy

Recognise quadratic form, solve by simple factorisation

② Medium

Quadratic formula, discriminant nature-of-roots MCQs

③ Hard

Find k for given root condition, word problems with two unknowns

★ PYQ

CBSE — D = 0 condition, word problem; NTSE — algebraic reasoning

Continue Your Preparation

📖 Chapter Notes Revise concepts & theory 🧠 MCQ Practice Topic-wise Questions 📘✔️ Exercises Step-by-step NCERT solutions ✔️❌ True / False Concept-based True & False ➡ Next Chapter Arithmetic Progressions

🎯 Knowledge Check

Maths — QUADRATIC EQUATIONS

50 Questions Class 10 MCQs

Which represents the standard form of a quadratic equation?

a a. $x^3+2x^2+3x+1=0$ b b. $2x^2-5x+7=0$ c c. $x+3=0$ d d. $x^4-x=0$

The discriminant of $x^2-4x+4=0$ is:

a a. 0 b b. 16 c c. -16 d d. 32

$D>0$ means roots are:

a a. Real and equal b b. Real and distinct c c. No real roots d d. Imaginary

Sum of roots for $ax^2+bx+c=0$ is:

a a. $c/a$ b b. $-b/a$ c c. $b/a$ d d. $-c/a$

Product of roots for $ax^2+bx+c=0$ is:

a a. $-b/a$ b b. $c/a$ c c. $b/a$ d d. $-c/a$

Roots of $x^2-5x+6=0$:

a a. 2, 3 b b. 1, 6 c c. -2, -3 d d. 5, 1

$2x^2+3x+1=0$ has how many real roots?

a a. 0 b b. 1 c c. 2 equal d d. 2 distinct

Equation with roots 2, -3:

a a. $x^2+x-6=0$ b b. $x^2-x+6=0$ c c. $x^2-x-6=0$ d d. $x^2+x+6=0$

$D=0$ implies:

a a. Distinct real roots b b. Equal real roots c c. No real roots d d. Positive roots

Nature of roots for $3x^2-6x+3=0$:

a a. Distinct real b b. Equal real c c. No real d d. Two positive

Which is quadratic?

a a. $x^3=0$ b b. $x^2=4$ c c. $x+1=0$ d d. $x^4=1$

Product of roots $4x^2+4x+1=0$:

a a. $1/4$ b b. $-1/4$ c c. 1 d d. 4

Sum of roots $x^2-7x+10=0$:

a a. 7 b b. -7 c c. 10 d d. -10

Which has $D<0$?

a a. $x^2+1=0$ b b. $x^2-1=0$ c c. $x^2=0$ d d. $x^2-2x+1=0$

Roots of $x^2+2x-3=0$:

a a. 1, -3 b b. -1, 3 c c. 2, -1.5 d d. -2, 1.5

Quadratic formula is:

a a. $\frac{-b\pm\sqrt{D}}{2a}$ b b. $\frac{b\pm\sqrt{D}}{2a}$ c c. $\frac{-b\pm D}{2a}$ d d. $\frac{b\pm D}{a}$

For $x^2-4=0$, roots:

a a. 2, -2 b b. 4, 0 c c. -4, 4 d d. 2, 2

If roots are equal, then:

a a. $D>0$ b b. $D=0$ c c. $D<0$ d d. $a=0$

$5x^2+5=0$ has:

a a. Two real roots b b. One real root c c. No real roots d d. Infinite roots

Sum for roots $1/2,3/2$:

a a. 1 b b. 2 c c. $3/4$ d d. $9/4$

Solve $x^2-7x+12=0$:

a a. 3, 4 b b. -3, -4 c c. 7, 12 d d. 1, 12

Using formula, $2x^2-3x-2=0$:

a a. 2, $-1/2$ b b. -2, $1/2$ c c. $3/2$, $-2/3$ d d. 1, -2

Verify if $x=2$ satisfies $x^2-4x+4=0$:

a a. Yes b b. No c c. Partially d d. Cannot say

Roots $x^2+5x+6=0$:

a a. -2, -3 b b. 2, 3 c c. -5, -6 d d. 1, 6

$D$ of $3x^2+6x+3=0$:

a a. 0 b b. 9 c c. -9 d d. 36

Solve $x^2+x-12=0$:

a a. 3, -4 b b. -3, 4 c c. 12, -1 d d. 1, 12

Product for $4x^2-8x+3=0$:

a a. $3/4$ b b. $-3/4$ c c. $8/4$ d d. 3

Nature $x^2+2x+2=0$:

a a. Real distinct b b. Real equal c c. No real d d. Positive

Equation roots 1, 1:

a a. $x^2-2x+1=0$ b b. $x^2+2x+1=0$ c c. $x^2-x+1=0$ d d. $x^2+x+1=0$

Solve $6x^2-7x-3=0$:

a a. $3/2$, $-1/2$ b b. $-3/2$, $1/2$ c c. $7/6$, 3 d d. 1, -3

$x^2-2x-8=0$ roots:

a a. 4, -2 b b. -4, 2 c c. 8, 0 d d. 2, -4

If one root 3, sum 7: other root?

a a. 4 b b. -4 c c. 21 d d. 3

Completing square $x^2+6x+8=0$:

a a. -4, -2 b b. 4, 2 c c. -3, -3 d d. 3, 3

$9x^2-6x+1=0$ roots:

a a. Equal b b. Distinct c c. Imaginary d d. Zero

Product roots zero implies:

a a. $c=0$ b b. $b=0$ c c. $a=0$ d d. $D=0$

Solve $x^2-9=0$:

a a. 3, -3 b b. 9, 0 c c. -9, 9 d d. 3, 3

$D$ for $x^2+4x+4=0$:

a a. 0 b b. 16 c c. -16 d d. 32

Roots $2x^2+5x+3=0$:

a a. -1, $-3/2$ b b. 1, $3/2$ c c. $-5/2$, -3 d d. 2, $3/5$

If $D=16$, $a=1$, $b=-8$: $c=$?

a a. 12 b b. -12 c c. 8 d d. 16

Verify $x=-3$ for $x^2+2x-3=0$:

a a. Yes b b. No c c. Zero d d. Positive

Rectangle length=width+2, area 15: equation?

a a. $x^2+2x-15=0$ b b. $x^2-2x+15=0$ c c. $x^2+2x+15=0$ d d. $2x^2-15=0$

Speed $x$ km/h, return $x+5$, time diff 1h for 120km:

a a. $x^2+5x-600=0$ b b. $x^2-5x-600=0$ c c. $x^2+10x-600=0$ d d. $x^2-10x+600=0$

Two numbers sum 10, product 21: equation?

a a. $x^2-10x+21=0$ b b. $x^2+10x-21=0$ c c. $x^2-21x+10=0$ d d. $10x^2+21=0$

Father 3×son age, 15yrs ago 5×: son now?

a a. 18 b b. 15 c c. 20 d d. 21

Boat speed $x$ upstream, downstream $x+10$, diff 24km in 4h:

a a. $x^2+10x-60=0$ b b. $x^2-10x-60=0$ c c. $2x^2-240=0$ d d. $x^2+20x-60=0$

Garden length=width+4, area=96: equation?

a a. $x^2+4x-96=0$ b b. $x^2-4x-96=0$ c c. $x^2+8x+96=0$ d d. $x(x+4)=96$

Projectile $h=10t-5t^2$ max at?

a a. $t=1$ b b. $t=2$ c c. $t=5$ d d. $t=10$

Numbers differ by 3, sum squares 61: larger?

a a. 7 b b. 5 c c. 8 d d. 4

Trains $x,x+2$ km/h, relative 80km/h covers 240km:

a a. $x^2+2x-1600=0$ b b. $x(x+2)=240$ c c. $2x^2=240$ d d. $x^2-2x-240=0$

Roots $\alpha,\beta$: $\alpha+\beta=5$, $\alpha\beta=6$, then $\alpha^2+\beta^2=$?

a a. 13 b b. 25 c c. 30 d d. 11

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Frequently Asked Questions

A quadratic equation is an equation of the form $ax^2 + bx + c = 0$ where $a,\ b\, c$ are real numbers and $a \neq 0$.

If $a = 0$, the equation becomes linear and no longer contains a squared term, so it cannot be quadratic.

The standard form is $ax^2 + bx + c = 0$.

The word “quadratic” comes from “quad,” meaning square, because the highest power of the variable is 2.

The solutions of $ax^2 + bx + c = 0$ are $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

The discriminant $D$ is the expression $b^2 - 4ac$ found inside the square root of the quadratic formula.

It indicates two distinct real roots.

It indicates one real and repeated root.

It indicates no real roots; the solutions are complex.

By splitting the middle term into two terms whose product is (ac), factoring the expression, and using the zero-product property.

If $pq = 0$, then either $p = 0$ or $q = 0$. It is used to solve factored quadratic equations.

It means expressing $bx$ as the sum of two terms whose product equals $ac$, helping in factorization.

It is a method of rewriting a quadratic as a perfect square expression to solve the equation.

It helps derive the quadratic formula and solve equations that are not easy to factor.

Ensure $a = 1$, take half of the coefficient of $x$, square it, add it to both sides, form a perfect square, and solve.

QUADRATIC EQUATIONS – Learning Resources

Detailed Notes

True / False

NCERT Solutions

ACADEMIA AETERNUM तमसो मा ज्योतिर्गमय · Est. 2025

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Quadratic Equations

MCQ Bank Snapshot

Why Practise These MCQs?

Topic-wise MCQ Breakdown

Must-Know Formulae Before You Start

MCQ Solving Strategy

⚠ Common Traps & Errors

Difficulty Ladder

Continue Your Preparation

Maths — QUADRATIC EQUATIONS

Frequently Asked Questions

Q 01 What is a quadratic equation?

Q 02 Why must the coefficient of \(x^2\) be non-zero?

Q 03 What is the standard form of a quadratic equation?

Q 04 What is the meaning of the term “quadratic”?

Q 05 What is the quadratic formula?

Q 06 What is the discriminant?

Q 07 What does \(D > 0\) indicate?

Q 08 What does \(D = 0\) indicate?

Q 09 What does \(D < 0\) indicate?

Q 10 How do you solve a quadratic equation by factorization?

Q 11 What is the zero-product property?

Q 12 What does “splitting the middle term” mean?

Q 13 What is the method of completing the square?

Q 14 Why is completing the square important?

Q 15 What are the steps of completing the square?

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QUADRATIC EQUATIONS – Learning Resources

Detailed Notes

True / False

NCERT Solutions

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