# Worksheet on Nature of the Roots of a Quadratic Equation

Practice the questions given in the Worksheet on nature of the roots of a quadratic equation.

We know the nature of the roots of a quadratic equation depends completely on the value of its discriminant.

1. Without solving, comment upon the nature of roots of each of the following equations:

(a) 7x$$^{2}$$ - 9x + 2 = 0

(b) 6x$$^{2}$$ - 13x + 4 = 0

(c) 25x$$^{2}$$ - 10x + 1 = 0

(d) x$$^{2}$$ + 2√3 x - 9 = 0

(e) x$$^{2}$$ - ax + b$$^{2}$$ = 0

(f) 2x$$^{2}$$ + 8x + 9 = 0

2. Find the discriminant of the following equations.

(a) x(x - 2) + 1 = 0

(b) $$\frac{1}{x + 2}$$ + $$\frac{1}{x - 2}$$ = 2

3. Prove that none of the following equations has any real solution.

(a) x$$^{2}$$ + x + 1 = 0

(b) x(x - 1) + 1 = 0

(c) x + $$\frac{4}{x}$$ - 1 = 0, x ≠ 0

(d) x(x + 1) + 3(x + 3) = 0

(e) $$\frac{x}{x + 1}$$ + $$\frac{3}{x - 1}$$ = 0; x ≠ 1, -1

4. Find the value of ‘p’, if the following quadratic equation has equal roots: 4x$$^{2}$$ - (p - 2)x + 1 = 0

5. Prove that each of the following equation has only one solution. Find the solution.

(a) 4y$$^{2}$$ - 28y + 49 = 0

(b) $$\frac{1}{4}$$x$$^{2}$$ + $$\frac{1}{3}$$x + $$\frac{1}{9}$$ = 0

(c) 8x(2x - 5) + 25 = 0

6. Find the value of λ for which the equation λx$$^{2}$$ + 2x + 1 = 0 has real and distinct roots.

7. For what value of k will each of the following equations give equal roots? Also, find the solution for that value of k.

(a) 3x$$^{2}$$ + kx + 2 = 0

(b) kx$$^{2}$$ - 4x + 1 = 0

(c) 5x$$^{2}$$ + 20x + k = 0

(d) (k - 12)x$$^{2}$$ + 2(k - 12)x + 2 = 0

8. The equation 3x$$^{2}$$ - 12x + z - 5 = 0 has equal roots. Find the value of z.

9. Find k for which the equation 4x$$^{2}$$ + kx + 9 = 0 will be satisfied by only one real value of x. Also find the solution.

10. Find the value of ‘z’, if the following equation has equal roots:

(z - 2)x$$^{2}$$ - (5 + z)x + 16 = 0

11. Find the nature of roots of the following equation. If they are real, find them.

(a) 3x$$^{2}$$ - 2x + $$\frac{1}{3}$$ = 0

(b) 3x$$^{2}$$ - 6x + 2 = 0

Answers for the Worksheet on nature of the roots of a quadratic equation are given below.

1. (a) Rational and unequal

(b) Irrational and unequal

(c) Rational (real) and equal

(d) Irrational and unequal (since, b = 2√3 is irrational)

(e) Irrational and unequal

(f) Imaginary roots

2. (a) 0

(b) 17

4. p = -2 or 6

5. (a) $$\frac{7}{2}$$

(b) -$$\frac{2}{3}$$

(c) $$\frac{5}{4}$$

6. All real values of λ < 1.

7. (a) ±2√6; when k = 2√6, solution = -$$\frac{2}{√6}$$ and when k = -2√6, solution = $$\frac{2}{√6}$$

(b) 4; solution = -$$\frac{1}{2}$$

(c) 20; solution = -2

(d) 14; solution = -1

8. z = 17

9. ± 12; when k = 12, solution = -$$\frac{3}{2}$$ and when k = -12, solution = $$\frac{3}{2}$$

10. z = 3 or 51

11. (a) Real, Roots = $$\frac{1}{3}$$, $$\frac{1}{3}$$

(b) Real, Roots = $$\frac{√3 - 1}{√3}$$, $$\frac{√3 + 1}{√3}$$

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