# 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 = $$\frac{√3 - 1}{√3}$$, $$\frac{√3 + 1}{√3}$$

Formation of Quadratic Equation in One Variable

Word Problems on Quadratic Equations by Factoring

Worksheet on Formation of Quadratic Equation in One Variable

Worksheet on Nature of the Roots of a Quadratic Equation

Worksheet on Word Problems on Quadratic Equations by Factoring

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Adding 1-Digit Number | Understand the Concept one Digit Number

Sep 17, 24 02:25 AM

Understand the concept of adding 1-digit number with the help of objects as well as numbers.

2. ### Counting Before, After and Between Numbers up to 10 | Number Counting

Sep 17, 24 01:47 AM

Counting before, after and between numbers up to 10 improves the child’s counting skills.

3. ### Worksheet on Three-digit Numbers | Write the Missing Numbers | Pattern

Sep 17, 24 12:10 AM

Practice the questions given in worksheet on three-digit numbers. The questions are based on writing the missing number in the correct order, patterns, 3-digit number in words, number names in figures…

4. ### Arranging Numbers | Ascending Order | Descending Order |Compare Digits

Sep 16, 24 11:24 PM

We know, while arranging numbers from the smallest number to the largest number, then the numbers are arranged in ascending order. Vice-versa while arranging numbers from the largest number to the sma…