# Roots of a Quadratic Equation | Discriminate of Quadratic Equation

We will discuss here the different cases of discriminant to understand the nature of the roots of a quadratic equation.

We know that the roots of the quadratic equation ax^{2}+bx+c=0 where aâ‰ 0 are given by the quadratic formula as:

x=-bÂ±âˆš(b^2-4ac/2a

We see that there are two possible values for x, as discriminated by the parts of the formula

Â±âˆš (b^2-4ac).

The nature of the roots of an equation depends on the value of the expression b^{2}-4ac, which is called its discrimination.

## Discriminate of Quadratic Equation

The expression (b^{2}-4ac) is called the discriminant of the quadratic equation.

Generally denoted by discriminant of the quadratic equation, D or âˆ†

Therefore,

Discriminant =âˆ†= b^{2}-4ac

We shall discuss the following cases about the nature of the roots of the quadratic equation ax^{2}+bx+c=0

When a,b, and c are real numbers,aâ‰ 0

## Case:1.When discriminant is equal to zero(b^{2}-4ac=0)

If the discriminant is equal to zero (b2-4ac=0), then the roots of a quadratic equation are real and repeated equally.

**For example:- **find the nature of the roots of the quadratic equation .9x^{2}-12x+4=0 by solving them

**Solution:** 9x^{2}-12x+4=0 comparing with ax^{2}+bx+c=0

We have, a=9, b=-12, c=4

Discriminant =âˆ†=b^{2}-4ac

Discriminant =âˆ†= (-12)^{2}-4(9)(4)

Discriminant =âˆ†= 144-144

Discriminant =âˆ†=0

** Clearly, the discriminant of the given quadratic equation is zero**.

Therefore, the roots of the given quadratic equation are real and equal.

Solve the equation:

9x^{2}-12x+4=0

9x^{2}-6x-6x+4=0 âˆ´ by using the factorization method

3x (3x-2)-2(3x-2) =0

(3x-2)(3x-2)=0

3x-2=0 or 3x-2=0

3x=2 3x=2

X=2/3 x=2/3

The given equation has two solutions: 2/3 and 2/3

Solution set ={2/3,2/3}

The solution of an equation is also called its roots.

** Clearly, the roots of a given quadratic equation are real and equal**.

## Case:2.When discriminant is less than zero or negative (b^{2}-4ac<0)

If the discriminant is negative (b2-4ac<0) thenâˆš(b^2-4ac will be imaginary so the roots of a quadratic equation are complex/imaginary and distinct/unequal.

**For example:** Discuss the nature of the roots of a quadratic equation.x^{2}+2x+3=0

Solution:

x^{2}+2x+3=0 comparing with ax^{2}+bx+c=0,we have

a=1 ,b=2,c=3

Discriminant =âˆ†=b^{2}-4ac

Discriminant =âˆ†=(2)^{2}-4(1)(3)

Discriminant =âˆ†=4-12

Discriminant =âˆ†=-8

Clearly, the discriminant of the given quadratic equation (2nd-degree polynomial) is negative.

Therefore, the roots of the given quadratic equation are imaginary and distinct/unequal.

Solve the equation:

x^{2}+2x+3=0

Comparing the standard form of quadratic equation

a=1,b=2,c=3

By using the quadratic formula

x=-bÂ±âˆš(b^2-4ac/2a

Put the value a, b and c

x=-(2)Â±âˆš(2^2-4(1)(3)/2(1)

x=-2Â±âˆš(4-12/2

x=-2Â±âˆš(-8/2

x=-2Â±âˆš(4X2 i/2

x=-2Â±2âˆš(2 i/2

x=2(-1Â±âˆš(2 i)/2

x=-1Â±âˆš2 i

*Clearly, the roots of the given equation are imaginary and unequal*

## Case:3. When discriminant is greater than zero or positive (b^{2}-4ac>0)

If the discriminant is positive (b2-4ac>0) then âˆš(b^2-4ac will be real .then the roots of a quadratic equation are real and distinct/unequal.

**For example:-** Discuss the nature of the roots of quadratic equation 2x^{2}+5x-1=0

Solution:

2x^{2}+5x-1=0 comparing with ax^{2}+bx+c=0 we have

a=2,b=5,c=-1

Discriminant =âˆ†=b^{2}-4ac

Discriminant =âˆ†= (5)^{2}-4(2)(-1)

Discriminant =âˆ†=25+8

Discriminant =âˆ†=33

Clearly, the discriminant of a given quadratic equation is positive.

Therefore, the roots of the given quadratic equation are real and distinct /unequal.

## Case:4. When discriminant is greater than zero and perfect square (b^{2}-4ac>0 )

If the discriminant is greater than zero (b2-4ac>0) or positive and perfect square then the roots of a quadratic equation are real, rational, and unequal.

**For example:** Discuss the nature of the roots of the quadratic equation (p+q) x^{2}-px-q=0

Solution:

(p+q) x^{2}-px-q=0 comparing with ax^{2}+bx+c=0 we have

a=(p+q) ,b=-p ,c=-q

Discriminant =âˆ†=b^{2}-4ac

Discriminant =âˆ†=(-p)^{2}-4(p+q)(-q)

Discriminant =âˆ†=p^{2}+4pq+4q^{2}

Discriminant =âˆ†=p^{2}+2(p)(2q)+(2q)^{2}

Discriminant =âˆ†= (p+q)^{ 2}

Which is a perfect square

*Hence the roots of a given quadratic equation are*

* Real, rational, and unequal*

**For example: x ^{2}+4x+3=0**

**Solution:**

x^{2}+4x+3=0 comparing with ax^{2}+bx+c=0 we have

a=1 ,b=4 ,c=3

Discriminant =âˆ†=b^{2}-4ac

Discriminant =âˆ†=(4)^{2}-4(1)(3)

Discriminant =âˆ†=16-12

Discriminant =âˆ†=4

Discriminant =âˆ†=(2)^{2} >0

*Which is a perfect square*

Solve the equation:

x^{2}+4x+3=0

By using the factorization method

X^{2}+3x+x+3=0

X(x+3)+1(x+3)=0

(x+3)(x+1)=0

X+3=0 or x+1=0

X=-3 x=-1

Solution set={-3,-1}

Clearly, the discriminant is given an equation greater than zero and a perfect square.

** Clearly,** the roots of the given equation are real, rational,

**.**

*and unequal*## Case 5:- When the discriminant is greater than zero and not a perfect square (b^{2}-4ac>0 )

If the discriminant is positive but not a perfect square then the roots of the quadratic equation are real, irrational,unequal

**For example:** Discuss the nature of the roots of quadratic equations.4x^{2}+6x+1=0

Solution:

4x^{2}+6x+1=0 comparing with ax^{2}+bx+c=0 we have

a=4 ,b=6 ,c=1

Discriminant =âˆ†=b^{2}-4ac

Discriminant =âˆ†= (6)^{2}– 4(4)(1)

Discriminant =âˆ†=36-16

Discriminant =âˆ†= 20>0 but not a perfect square

Clearly, if the discriminant of the given quadratic equation is greater than zero but not perfect square the roots of the quadratic equation are real, unequal, and irrational.

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