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Quadratic Equations Quiz - CBSE Class 10 Mathematics

Free practice quiz: Quadratic Equations (CBSE Class 10 Mathematics). 15 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This CBSE Class 10 Mathematics practice quiz has 15 multiple-choice questions on Quadratic Equations, and 15 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

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About this topic

A quadratic equation has degree 2 and at most two real roots. Solve it by factorisation, completing the square, or the quadratic formula. The discriminant tells you the nature of the roots without solving.

Standard form: ax² + bx + c = 0, where a ≠ 0.
Roots (zeros) are the values of x that satisfy the equation.
Factorisation: split the middle term so that sum = b and product = ac.
Quadratic formula works for every quadratic equation.
Discriminant D = b² − 4ac decides the nature of roots.
If D < 0, there are no real roots.
Always rewrite the equation in standard form before solving.

Learn the key ideas Beta

Standard Form and Factorisation

A quadratic equation can be written as ax^2 + bx + c = 0, where a is not zero.
To factorise, split bx into two terms whose product is ac and whose sum is b.
If a product of two factors is zero, then at least one factor must be zero.

Standard form

\[ax^2 + bx + c = 0,\; a \neq 0\]

Zero product rule

\[(x-p)(x-q)=0 \Rightarrow x=p \text{ or } x=q\]

Difference of squares

\[x^2 - m^2 = (x-m)(x+m)\]

Common mistake: Dividing both sides by x loses the root x = 0. Take x common as a factor instead.

The Quadratic Formula

The quadratic formula works for every quadratic, even when factorising is hard.
First write the equation in standard form, then read off a, b and c with their signs.

Quadratic formula

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Common mistake: Forgetting that -b changes the sign of b, or dividing only the square root part by 2a.

Discriminant and Nature of Roots

The discriminant D = b^2 - 4ac tells the type of roots without solving.
For equal roots, set D = 0 and solve for the unknown constant.

Discriminant

\[D = b^2 - 4ac\]

Two distinct real roots

\[D > 0\]

Two equal real roots

\[D = 0 \Rightarrow x = -\frac{b}{2a}\]

No real roots

\[D < 0\]

Common mistake: Saying D < 0 means no roots at all. It means no real roots.

Sum and Product of Roots

If α and β are the roots, their sum and product come directly from a, b and c.
Rewrite expressions like α^2 + β^2 in terms of the sum and product, without finding the roots.

Sum of roots

\[\alpha + \beta = -\frac{b}{a}\]

Product of roots

\[\alpha \beta = \frac{c}{a}\]

Sum of squares

\[\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\]

Square of difference

\[(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta\]

Common mistake: Writing the sum of roots as b/a. The correct sum is -b/a, so watch the minus sign.

Forming Equations from Roots

If you know both roots, you can build the quadratic from their sum and product.
Reciprocal roots have product one. Negative reciprocal roots have product minus one.

Equation from roots

\[x^2 - (\alpha + \beta)x + \alpha\beta = 0\]

Ratio expression

\[\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta}\]

Consecutive integer roots

\[\alpha = n,\; \beta = n+1 \Rightarrow (\alpha - \beta)^2 = 1\]

Common mistake: Writing x^2 + (α + β)x + αβ = 0. The middle term must have a minus sign.

Exam tips

Always rearrange the equation into standard form before reading off a, b and c.
Check your roots by substituting them back into the original equation.
For nature-of-roots questions, compute only D. You do not need to solve the equation.

Beta: made with AI and checked by a second AI. A teacher has not reviewed it yet.

Sample questions

What is the value of x in the equation x^2 - 5x + 6 = 0?
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Answer: B. 2 and 3
The equation x^2 - 5x + 6 = 0 can be factored into (x - 2)(x - 3) = 0, giving solutions x = 2 and x = 3.
Solve the equation x^2 - 9 = 0.
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Answer: A. 3 and -3
The equation x^2 - 9 = 0 can be rewritten as (x - 3)(x + 3) = 0, so the solutions are x = 3 and x = -3.
Determine the value of k for which the equation x^2 + kx + 4 = 0 has equal roots.
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Answer: C. -4
For equal roots, the discriminant (b^2 - 4ac) must be zero. Here, a = 1, b = k, c = 4. k^2 - 16 = 0 implies k = ±4. Choose -4 as per given options.
Find the nature of the roots for the quadratic equation 2x^2 - 6x + 3 = 0.
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Answer: A. Real and distinct
The discriminant is b^2 - 4ac = 36 - 24 = 12, which is greater than zero, indicating real and distinct roots.
If α and β are the roots of x^2 + 4x + 8 = 0, find the value of α/β + β/α.
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Answer: B. 0
α/β + β/α = (α^2 + β^2) / αβ. Using known α + β and αβ, calculate α^2 + β^2 and proceed as explained previously.