Important Questions for NCERT Class 10 Maths Chapter 4 – Quadratic Equations

Quadratic Equations in Class 10 Maths. If you’re gearing up for the CBSE board exam 2024-2025, these will be handy to practice and aim for good scores. These questions are aligned with the NCERT book, designed by experts who’ve carefully studied the exam pattern. Expect similar questions in your Math paper.

These questions focus on quadratic equations and how to solve them using factorization methods to find their roots. By working through these, you’ll solidify your understanding and be better prepared for your exam. We’ve included 15 questions along with detailed solutions below. Make sure to go over them to strengthen your Math skills. Additionally, tackling extra practice questions will further enhance your problem-solving abilities in quadratic equations.

Important Questions for Class 10 Maths Chapter 4: Quadratic Equations

Question and Answer for Class 10 Maths Chapter 4: Quadratic Equations (1Mark)

Question: Define a quadratic equation.

Answer: A quadratic equation is a polynomial equation of the second degree, where the highest power of the variable is 2.

Question: What is the standard form of a quadratic equation?

Answer: The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Question: State the quadratic formula.

Answer: The quadratic formula is x = (-b ± √(b^2 – 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.

Question: What does the discriminant of a quadratic equation determine?

Answer: The discriminant of a quadratic equation (Δ = b^2 – 4ac) determines the nature of the roots of the equation. If Δ > 0, the equation has two distinct real roots. If Δ = 0, the equation has two equal real roots. If Δ < 0, the equation has two complex (non-real) roots.

Question: How many solutions can a quadratic equation have?

Answer: A quadratic equation can have either two real solutions, one real solution (in case of repeated roots), or two complex solutions.

Question and Answer for Class 10 Maths Chapter 4: Quadratic Equations (3 Mark)

Question: Solve

$2{𝑥}^2+3𝑥-5=0$

2x2+3x−5=0 using the quadratic formula.

Answer:

$𝑥=\frac{-3\pm \sqrt{3^2-4\cdot 2\cdot (-5)}}{2\cdot 2}$

x=2⋅2−3±32−4⋅2⋅(−5)​​

Question: Factorize

${𝑥}^2+5𝑥+6$

x2+5x+6.

Answer:

${𝑥}^2+5𝑥+6=(𝑥+2)(𝑥+3)$

x2+5x+6=(x+2)(x+3)

Question: Find the roots of

$3{𝑥}^2-4𝑥-4=0$

3x2−4x−4=0 using the factorization method.

Answer:

$𝑥=\frac{2}{3},-2$

x=32​,−2

Question: If

$𝑥=-2$

x=−2 is a root of the quadratic equation

$2{𝑥}^2+𝑘𝑥-15=0$

2x2+kx−15=0, find the value of

$𝑘$

k.

Answer:

$𝑘=-1$

k=−1

Question: Solve the quadratic equation

$4{𝑥}^2+4𝑥+1=0$

4x2+4x+1=0 by completing the square method.

Answer:

$𝑥=-\frac{1}{2}$

x=−21​

Question: Determine the discriminant of the quadratic equation

$3{𝑥}^2-5𝑥+2=0$

3x2−5x+2=0 and state the nature of its roots.

Answer: Discriminant (

$𝐷$

D) =

$5^2-4\cdot 3\cdot 2=1$

52−4⋅3⋅2=1, since

$𝐷>0$

D>0, the roots are real and distinct.

Question: If one root of the quadratic equation

${𝑥}^2+𝑚𝑥-15=0$

x2+mx−15=0 is 3, find the value of

$𝑚$

m.

Answer: By using sum and product of roots,

$𝑚=-12$

m=−12.

Question: Find the roots of the quadratic equation

${𝑥}^2+8𝑥+15=0$

x2+8x+15=0 by the quadratic formula method.

Answer:

$𝑥=-5,-3$

x=−5,−3

Question: Determine the value of

$𝑘$

k for which the equation

${𝑥}^2-2𝑘𝑥+9=0$

x2−2kx+9=0 has equal roots.

Answer:

$𝑘=3$

k=3

Question: Factorize

$3{𝑥}^2+7𝑥-6$

3x2+7x−6.

Answer:

$3{𝑥}^2+7𝑥-6=(3𝑥-2)(𝑥+3)$

3x2+7x−6=(3x−2)(x+3)

Question: Solve the equation

$4{𝑥}^2-12𝑥+9=0$

4x2−12x+9=0 using the method of splitting the middle term.

Answer:

$𝑥=\frac{3}{2}$

x=23​

Question: Find the roots of

$2{𝑥}^2-7𝑥+3=0$

2x2−7x+3=0 by using the quadratic formula.

Answer:

$𝑥=\frac{7\pm \sqrt{7^2-4\cdot 2\cdot 3}}{2\cdot 2}$

x=2⋅27±72−4⋅2⋅3​​

Question: If one root of

$𝑎{𝑥}^2-5𝑥+6=0$

ax2−5x+6=0 is 2, find the value of

$𝑎$

a.

Answer:

$𝑎=\frac{1}{2}$

a=21​

Question: Determine the roots of

$3{𝑥}^2-2\sqrt{6}𝑥+1=0$

3x2−26​x+1=0 by using the quadratic formula.

Answer:

$𝑥=\frac{\sqrt{6}\pm \sqrt{6}}{3}$

x=36​±6​​

Question: Factorize

$2{𝑥}^2-5𝑥-3$

2x2−5x−3.

Answer:

$2{𝑥}^2-5𝑥-3=(2𝑥+1)(𝑥-3)$

2x2−5x−3=(2x+1)(x−3)

Question and Answer for Class 10 Maths Chapter 4: Quadratic Equations (5 Mark)

Question: Solve the quadratic equation 2x^2 – 5x – 3 = 0.

Answer: The solutions are x = 3/2 and x = -1.

Question: Find the roots of the quadratic equation x^2 + 4x + 4 = 0.

Answer: The only root is x = -2.

Question: Determine the values of k for which the quadratic equation (k+1)x^2 – 4(k+1)x + 4 = 0 has equal roots.

Answer: For equal roots, the discriminant must be zero. So, (4(k+1))^2 – 4(k+1)(k+1) = 0. Solving this equation gives k = 1 or k = -1.

Question: If one root of the quadratic equation x^2 – px + 12 = 0 is 4, find the value of p.

Answer: Since 4 is a root, the quadratic equation becomes (x – 4)(x – ?) = 0. Equating coefficients with the original equation, we get p = 16.

Question: Solve the quadratic equation 3x^2 – 2x – 1 = 0 by using the quadratic formula.

Answer: Using the quadratic formula, we find x = (2 ± √16) / 6. So, x = (2 ± 4) / 6. This gives two solutions: x = 1 and x = -1/3.

Question: Factorize the quadratic expression x^2 + 7x + 12.

Answer: The expression factors as (x + 3)(x + 4).

Question: Solve the quadratic equation 5x^2 + 6x – 2 = 0 using the method of completing the square.

Answer: Completing the square, we get 5(x + 3/5)^2 – 49/5 = 0. Rearranging, we find (x + 3/5)^2 = 49/25. Taking the square root, we obtain x + 3/5 = ±7/5. So, x = 2/5 or x = -10/5.

Question: Determine the nature of the roots of the quadratic equation 4x^2 + 4x + 1 = 0.

Answer: Since the discriminant is zero, the roots are real and equal.

Question: Find the roots of the quadratic equation 2x^2 – 7x + 3 = 0 by factorization.

Answer: Factoring, we get (2x – 1)(x – 3) = 0. So, the roots are x = 1/2 and x = 3.

Question: If the sum of the roots of the quadratic equation x^2 – px + q = 0 is 7 and one of the roots is 3, find the value of q.

Answer: Since the sum of the roots is 7 and one root is 3, the other root is 4. Therefore, q = 3 * 4 = 12.