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.

 

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

Solving Quadratic Equations

General Properties of Quadratic Equation

Methods of Solving Quadratic Equations

Roots of a Quadratic Equation

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

From Worksheet on Nature of the Roots of a Quadratic Equation to HOME PAGE




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.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. Shifting of Digits in a Number |Exchanging the Digits to Another Place

    May 19, 24 06:35 PM

    Shifting of Digits in a Number
    What is the Effect of shifting of digits in a number? Let us observe two numbers 1528 and 5182. We see that the digits are the same, but places are different in these two numbers. Thus, if the digits…

    Read More

  2. Formation of Greatest and Smallest Numbers | Arranging the Numbers

    May 19, 24 03:36 PM

    Formation of Greatest and Smallest Numbers
    the greatest number is formed by arranging the given digits in descending order and the smallest number by arranging them in ascending order. The position of the digit at the extreme left of a number…

    Read More

  3. Formation of Numbers with the Given Digits |Making Numbers with Digits

    May 19, 24 03:19 PM

    In formation of numbers with the given digits we may say that a number is an arranged group of digits. Numbers may be formed with or without the repetition of digits.

    Read More

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

    May 19, 24 02:23 PM

    Arranging Numbers
    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…

    Read More

  5. Comparison of Numbers | Compare Numbers Rules | Examples of Comparison

    May 19, 24 01:26 PM

    Rules for Comparison of Numbers
    Rule I: We know that a number with more digits is always greater than the number with less number of digits. Rule II: When the two numbers have the same number of digits, we start comparing the digits…

    Read More