# Important Questions for Chapter 4 - Quadratic Equations

With the right kind of attention and preparation, you can achieve good scores in math every time. With such an exhaustive list of questions contains step-by-step explanation and solutions for each question is also provided for your clear understanding and learning of the concepts. Get your set of questions now!

Important questions based on NCERT syllabus for Chapter 4 - Quadratic Equations:

Question 1: If one root of the quadratic equation 2x^2 + kx − 6 = 0 is 2, find the value of k. Also, find the other root.
Solution:
Since x = 2 is a root of the equation
2x^2 + kx − 6 = 0
∴2×2^2 + 2k − 6 =0
⇒ 8 + 2k − 6 = 0
⇒ 2k + 2 = 0
⇒ k = −1
Substituting k=−1 in the equation 2x^2 + kx − 6 = 0, we get
2x^2 − x − 6 = 0
⇒ 2x^2 − 4x + 3x − 6 = 0
⇒ 2x(x−2) + 3(x−2) = 0
⇒ (x−2)(2x+3) = 0
⇒ x−2 = 0,2x + 3 = 0
⇒ x = 2, x = −3/2
Hence, the other root is −3/2

p^2 x^2 + (p^2 − q^2)x − q^2 = 0, p ≠ 0
Solution:
We have, p^2 x^2 + (p^2 − q^2)x − q^2 = 0,p = 0
Comparing this equation with ax2+bx+c=0, we have
a = p^2, b = p^2 − q^2 and c = −q^2
∴ D = b^2 − 4ac = (p^2 − q^2)^2 − 4 × p^2 × −q^2
⇒ D = (p^2 − q^2)^2 + 4p^2q^2
⇒ D = (p^2 + q^2)^2
⇒ D > 0
So, the given equation has real roots given by
α = −b+√D/2a = −(p^2 − q^2) + (p^2 + q^2)2p^2 = q^2p^2
and, β = −b−√D/2a = −(p^2 − q^2) − (p^2 + q^2)/2p^2 = −1

Question 3: Seven years ago Varun's age was five times the square of Swati's age. Three years hence Swati's age will be two fifth of Varun's age. Find their present ages.
Solution:
Seven years ago, let Swati's age be x years, Varun's present age =(5x2+7) years
Three years hence, we have
Swati's age =(x+7+3) years=(x+10) years
Varun's age =(5x2+7+3) years=(5x2+10) years
It is given that three years hence Swati's age will be 25 of Varun's age.
∴ x + 10 = 25(5x^2 + 10)
⇒ x + 10 = 2x^2 + 4
⇒ 2x^2 − x − 6 = 0
⇒ 2x^2 − 4x + 3x − 6 = 0
⇒ 2x(x−2) + 3(x−2) = 0
⇒ (2x+3)(x−2) = 0
⇒ x − 2 = 0 [∵2x+3≠0 as x>0]
⇒ x = 2
Hence, Swati's pressent age = (2+7) years = 9 years
Varun's present age =(5×22+7) years = 27 years.