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Solving Quadratic Equations Quiz - CBSE Class 10 Mathematics

Free practice quiz: Solving Quadratic Equations (CBSE Class 10 Mathematics). 12 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This CBSE Class 10 Mathematics practice quiz has 12 multiple-choice questions on Solving Quadratic Equations. 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. We solve it by factorisation, completing the square, or the quadratic formula. The discriminant tells whether roots are real or not, before we solve.

Standard form: ax² + bx + c = 0, where a ≠ 0.
Roots (zeros) are the values of x that make the equation true.
Factorisation: split the middle term so that product = ac, sum = b.
Completing the square rewrites the equation as (x + p)² = q.
Quadratic formula works for every quadratic equation.
Discriminant D = b² − 4ac decides the nature of roots.
D > 0: two distinct real roots; D = 0: two equal real roots; D < 0: no real roots.

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Standard Form and Factorisation

A quadratic equation has the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
To factorise, split the middle term bx into two terms whose coefficients multiply to ac and add to b.
If a product of two factors is zero, at least one factor is zero, which gives the roots.

Standard form

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

Zero product rule

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

Perfect square pattern

\[x^2 + 2px + p^2 = (x + p)^2\]

Common mistake: Trying to factorise before moving every term to one side so that the other side equals zero.

Completing the Square, Vertex Form

Add and subtract the square of half the coefficient of x to create a perfect square.
Vertex form y = a(x − h)² + k shows the turning point (h, k) of the parabola directly.

Completing the square

\[x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2\]

Vertex form

\[y = a(x - h)^2 + k\]

Vertex

\[(h, k)\]

Common mistake: Getting the sign of h wrong: in a(x − h)², a bracket (x + m) means h = −m, not h = m.

Quadratic Formula and Discriminant

The quadratic formula solves every quadratic equation, including those that are hard to factorise.
The discriminant D = b² − 4ac decides the nature of the roots without solving the equation.
The number of real roots equals the number of points where the graph meets the x-axis.

Quadratic formula

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

Discriminant

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

Nature of roots

\[D > 0: \text{two distinct real roots};\ D = 0: \text{two equal real roots};\ D < 0: \text{no real roots}\]

Equal roots

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

Common mistake: Substituting a negative b or c without brackets, which gives the wrong sign for b² or −4ac.

Sum and Product of Roots

If α and β are the roots of ax² + bx + c = 0, their sum and product depend only on a, b, c.
Changing b changes the sum of the roots and the discriminant, so the roots and their nature can change.

Sum of roots

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

Product of roots

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

Equation from roots

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

Common mistake: Writing the sum of the roots as b/a instead of −b/a, which reverses the sign of the answer.

Exam tips

For 'one solution' or 'equal roots' questions, set b² − 4ac = 0 and solve for the unknown.
Divide every term by a common factor first; this makes factorising and calculating D easier.
Check your roots by substituting them back into the original equation.

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

Sample questions

What is the solution to the quadratic equation x² - 5x + 6 = 0?
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Answer: A. x = 2, 3
What is the sum of the roots of the equation x² - 8x + 16 = 0?
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Answer: B. 8
What is the discriminant of the equation 3x² + 12x + 9 = 0?
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Answer: A. 0
In the quadratic equation ax² + bx + c = 0, if a = 2, b = 8, and c = 10, what are the nature of the roots?
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Answer: C. Complex
Which of the following statements is true about the roots of the equation x² + 2x + 5 = 0?
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Answer: B. They are complex