Examine the Roots of a Quadratic Equation

Examining the roots of a quadratic equation means to see the type of its roots i.e., whether they are real or imaginary, rational or irrational, equal or unequal.

The nature of the roots of a quadratic equation depends entirely on the value of its discriminant b$^{2}$ - 4ac.

In a quadratic equation ax$^{2}$ + bx + c = 0, a ≠ 0 the coefficients a, b and c are real. We know, the roots (solution) of the equation ax$^{2}$ + bx + c = 0 are given by x = $\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$.

1. If b$^{2}$ - 4ac = 0 then the roots will be x = $\frac{-b ± 0}{2a}$ = $\frac{-b - 0}{2a}$, $\frac{-b + 0}{2a}$ = $\frac{-b}{2a}$, $\frac{-b}{2a}$.

Clearly, $\frac{-b}{2a}$ is a real number because b and a are real.

Thus, the roots of the equation ax$^{2}$ + bx + c = 0 are real and equal if b$^{2}$ – 4ac = 0.

2. If b$^{2}$ - 4ac > 0 then $\sqrt{b^{2} - 4ac}$ will be real and non-zero. As a result, the roots of the equation ax$^{2}$ + bx + c = 0 will be real and unequal (distinct) if b$^{2}$ - 4ac > 0.

3. If b$^{2}$ - 4ac < 0, then $\sqrt{b^{2} - 4ac}$ will not be real because $(\sqrt{b^{2} - 4ac})^{2}$ = b$^{2}$ - 4ac < 0 and square of a real number always positive.

Thus, the roots of the equation ax$^{2}$ + bx + c = 0 are not real if b$^{2}$ - 4ac < 0.

As the value of b$^{2}$ - 4ac determines the nature of roots (solution), b$^{2}$ - 4ac is called the discriminant of the quadratic equation.

Definition of discriminant: For the quadratic equation ax$^{2}$ + bx + c =0, a ≠ 0; the expression b$^{2}$ - 4ac is called discriminant and is, in general, denoted by the letter ‘D’.

Thus, discriminant D = b$^{2}$ - 4ac

Note:

When a quadratic equation has two real and equal roots we say that the equation has only one real solution.

Solved examples to examine the nature of roots of a quadratic equation:

1. Prove that the equation 3x$^{2}$ + 4x + 6 = 0 has no real roots.

Solution:

Here, a = 3, b = 4, c = 6.

So, the discriminant = b$^{2}$ - 4ac

= 4$^{2}$ - 4 ∙ 3 ∙ 6 = 36 - 72 = -56 < 0.

Therefore, the roots of the given equation are not real.

2. Find the value of ‘p’, if the roots of the following quadratic equation are equal (p - 3)x$^{2}$ + 6x + 9 = 0.

Solution:

For the equation (p - 3)x$^{2}$ + 6x + 9 = 0;

a = p - 3, b = 6 and c = 9.

Since, the roots are equal

Therefore, b$^{2}$ - 4ac = 0

⟹ (6)$^{2}$ - 4(p - 3) × 9 = 0

⟹ 36 - 36p + 108 = 0

⟹ 144 - 36p = 0

⟹ -36p = - 144

⟹ p = $\frac{-144}{-36}$

⟹ p = 4

Therefore, the value of p = 4.

3. Without solving the equation 6x$^{2}$ - 7x + 2 = 0, discuss the nature of its roots.

Solution:

Comparing 6x$^{2}$ - 7x + 2 = 0 with ax$^{2}$ + bx + c = 0 we have a = 6, b = -7, c = 2.

Therefore, discriminant = b$^{2}$ – 4ac = (-7)$^{2}$ - 4 ∙ 6 ∙ 2 = 49 - 48 = 1 > 0.

Therefore, the roots (solution) are real and unequal.

Note: Let a, b and c be rational numbers in the equation ax$^{2}$ + bx + c = 0 and its discriminant b$^{2}$ - 4ac > 0.

If b$^{2}$ - 4ac is a perfect square of a rational number then $\sqrt{b^{2} - 4ac}$ will be a rational number. So, the solutions x = $\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$ will be rational numbers. But if b$^{2}$ – 4ac is not a perfect square then $\sqrt{b^{2} - 4ac}$ will be an irrational numberand as a result the solutions x = $\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$ will be irrational numbers. In the above example we found that the discriminant b$^{2}$ – 4ac = 1 > 0 and 1 is a perfect square (1)$^{2}$. Also 6, -7 and 2 are rational numbers. So, the roots of 6x$^{2}$ – 7x + 2 = 0 are rational and unequal numbers.

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

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