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.
Answers:
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 = \(\frac{√3 - 1}{√3}\), \(\frac{√3 + 1}{√3}\)
Quadratic Equation
Introduction to Quadratic Equation
Formation of Quadratic Equation in One Variable
General Properties of Quadratic Equation
Methods of Solving Quadratic Equations
Examine the Roots of a Quadratic Equation
Problems on Quadratic Equations
Quadratic Equations by Factoring
Word Problems Using Quadratic Formula
Examples on Quadratic Equations
Word Problems on Quadratic Equations by Factoring
Worksheet on Formation of Quadratic Equation in One Variable
Worksheet on Quadratic Formula
Worksheet on Nature of the Roots of a Quadratic Equation
Worksheet on Word Problems on Quadratic Equations by Factoring
9th Grade Math
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