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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 b2 - 4ac.

In a quadratic equation ax2 + bx + c = 0, a β‰  0 the coefficients a, b and c are real. We know, the roots (solution) of the equation ax2 + bx + c = 0 are given by x = βˆ’b±√b2βˆ’4ac2a.

1. If b2 - 4ac = 0 then the roots will be x = βˆ’bΒ±02a = βˆ’bβˆ’02a, βˆ’b+02a = βˆ’b2a, βˆ’b2a.

Clearly, βˆ’b2a is a real number because b and a are real.

Thus, the roots of the equation ax2 + bx + c = 0 are real and equal if b2 – 4ac = 0.


2. If b2 - 4ac > 0 then √b2βˆ’4ac will be real and non-zero. As a result, the roots of the equation ax2 + bx + c = 0 will be real and unequal (distinct) if b2 - 4ac > 0.

3. If b2 - 4ac < 0, then √b2βˆ’4ac will not be real because (√b2βˆ’4ac)2 = b2 - 4ac < 0 and square of a real number always positive.

Thus, the roots of the equation ax2 + bx + c = 0 are not real if b2 - 4ac < 0.

As the value of b2 - 4ac determines the nature of roots (solution), b2 - 4ac is called the discriminant of the quadratic equation.


Definition of discriminant: For the quadratic equation ax2 + bx + c =0, a β‰  0; the expression b2 - 4ac is called discriminant and is, in general, denoted by the letter β€˜D’.

Thus, discriminant D = b2 - 4ac

Note:

Discriminant of

ax2 + bx + c = 0

Nature of roots of

ax2 + bx + c = 0

Value of the roots of

ax2 + bx + c = 0

b2 - 4ac = 0

Real and equal

- b2a, -b2a

b2 - 4ac > 0

Real and unequal

βˆ’b±√b2βˆ’4ac2a

b2 - 4ac < 0

Not real

No real value

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 3x2 + 4x + 6 = 0 has no real roots.

Solution:

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

So, the discriminant = b2 - 4ac

= 42 - 4 βˆ™ 3 βˆ™ 6 = 36 - 72 = -56 < 0.

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



2. Find the value of β€˜p’, if the roots of the following quadratic equation are equal (p - 3)x2 + 6x + 9 = 0.

Solution:

For the equation (p - 3)x2 + 6x + 9 = 0;

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

Since, the roots are equal

Therefore, b2 - 4ac = 0

⟹ (6)2 - 4(p - 3) Γ— 9 = 0

⟹ 36 - 36p + 108 = 0

⟹ 144 - 36p = 0

⟹ -36p = - 144

⟹ p = βˆ’144βˆ’36

⟹ p = 4

Therefore, the value of p = 4.


3. Without solving the equation 6x2 - 7x + 2 = 0, discuss the nature of its roots.

Solution:

Comparing 6x2 - 7x + 2 = 0 with ax2 + bx + c = 0 we have a = 6, b = -7, c = 2.

Therefore, discriminant = b2 – 4ac = (-7)2 - 4 βˆ™ 6 βˆ™ 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 ax2 + bx + c = 0 and its discriminant b2 - 4ac > 0.

If b2 - 4ac is a perfect square of a rational number then √b2βˆ’4ac will be a rational number. So, the solutions x = βˆ’b±√b2βˆ’4ac2a will be rational numbers. But if b2 – 4ac is not a perfect square then √b2βˆ’4ac will be an irrational numberand as a result the solutions x = βˆ’b±√b2βˆ’4ac2a will be irrational numbers. In the above example we found that the discriminant b2 – 4ac = 1 > 0 and 1 is a perfect square (1)2. Also 6, -7 and 2 are rational numbers. So, the roots of 6x2 – 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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