Extra Questions for Class 10 Maths Chapter 4 Quadratic Equations Based on Nature of Roots

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Extra Questions for Class 10 Maths Chapter 4 Quadratic Equations

Nature of Roots of a Quadratic Equation

Let the given equation be ax2 + bx + c = 0, where a = 0.

Then, the discriminant is given by

D = (b2 – 4ac) . And, the roots of the given equation are

Case 1: If D > 0, then the roots are real and distinct.

These roots are given by,

Case 2: If D = 0, then In this case, the roots are real and equal.

Each root = -b/2a

Case 3: If D < 0, then the roots are imaginary. So, the given equation has no real roots.

Question: Show that the equation 2×2 – 6x + 3 = 0 has real roots.

Solution:

The given equation is 2x2 – 6x + 3 = 0.
This is of the form ax2 + bx + c = 0, where a = 2, b = -6 and c = 3.
Therefore, D = (b2 – 4ac) = [(-6)2 – 4 x 2 x 3] = (36 – 24) = 12 > 0.

So, the given equation has real unequal roots.

Question: Find the value of k for which the roots of the quadratic equation kx(x – 2) + 6 = 0 are equal.

Solution:

The given equation is kx2 – 2kx + 6 = 0.
This is of the form ax2 + bx + c = 0, where a = k, b = -2k and c = 6.
D = (b2 – 4ac) = (4k2 – 4 x k x 6) = (4k2 – 24k).
For equal roots, we must have
D = 0

⇒ 4k2 – 24k=0

⇒ 4k(k – 6) = 0

Either k = 0 or k = 6.
Now, k ≠ 0, we get 6 = 0, which is not possible.
Therefore, k = 6.

Q.1. Show that the roots of the equation x2 + px – q2 = 0 are real for all real values of p and q.

Q.2. Find the nature of the roots of the following quadratic equations:
(i) 2x2 – 8x + 5 = 0 (iv) 5x(x – 2) + 6 = 0 (v) 12x2 – 4√15 x + 5 = 0

Ans. (i) Real and unequal (iv) Not real (v) Real and equal

Q.3. For what value of k are the roots of the quadratic equation kx(x – 2√5) + 10 = 0 real and equal?

Ans. k = 0 or k = 2

Q.4. If the quadratic equation (1 + m2)x2 + 2mcx + c2 – a2 = 0 has equal roots, prove that c2 = a2 (1 + m2) .

Q.5. (i) Find the values of k for which the quadratic equation (3k + 1)x2 + 2(k + 1)x + 1 = 0 has real and equal roots.
(ii) Find the value of k for which the equation x2 + k(2x + k – 1) + 2 = 0 has real and equal roots.

Ans. (i) k = 0 or k = 1 (ii) k = 2

Q.6. If –5 is a root of the quadratic equation 2x2 + px – 15 = 0 and the quadratic equation p(x2 + x) + k = 0 has equal roots, fi nd the value of k.

Ans. 49/28

Q.7. For what values of k are the roots of the quadratic equation 3x2 + 2kx + 27 = 0 real and equal?

Ans. k = 9 or k = -9

Q.8. If the roots of the equations ax2 + 2bx + c = 0 and bx2 – 2√ac x + b = 0 are simultaneously real then prove that b2 = ac.

Q.9. If –4 is a root of the equation x2 + 2x + 4p = 0, fi nd the value of k for which the quadratic equation x2 + px(1 + 3k) + 7(3 + 2k) = 0 has equal roots.

Ans. k=2 or k=-10/9

Q.10. Find the value of α for which the equation (α – 12)x2 + 2(α – 12)x + 2 = 0 has equal roots.

Ans. α=14

Q.11. Find the values of k for which the given quadratic equation has real and distinct roots:
(i) kx2 + 6x + 1 = 0
(ii) x2 – kx + 9 = 0
(iii) 9x2 + 3kx + 4 = 0

Ans. (i) k < 9 (ii) k > 6 or k < –6 (iii) k > 4 or k < –4

Q.12. For what values of p are the roots of the equation 4×2 + px + 3 = 0 real and equal?

Ans. p = 4√3 or p = -4√3

Frequently Asked Questions – FAQs

How to find roots of a quadratic equation?

There are various method to find out the roots of quadratic equation. You can use quadratic formula to find the roots of a quadratic equation.

What is the nature of root when D=0?

The quadratic equation has only one real root (or two equal roots -b/2a and -b/2a) when D or b2 – 4ac = 0.

What is the nature of roots when D<0?

The quadratic equation has two complex roots when b2 – 4ac < 0. The complex roots always occur in pairs.

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