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 ax²+bx+c=0 where a≠0 are given by the quadratic formula as:

x=-b±√(b^2-4ac/2a

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

±√ (b^2-4ac).

The nature of the roots of an equation depends on the value of the expression b²-4ac, which is called its discrimination.

Discriminate of Quadratic Equation

The expression (b²-4ac) is called the discriminant of the quadratic equation.

Generally denoted by discriminant of the quadratic equation, D or ∆

Therefore,

Discriminant =∆= b²-4ac

We shall discuss the following cases about the nature of the roots of the quadratic equation ax²+bx+c=0

When a,b, and c are real numbers,a≠0

Case:1.When discriminant is equal to zero(b²-4ac=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 .9x²-12x+4=0 by solving them

Solution: 9x²-12x+4=0 comparing with ax²+bx+c=0

 We have, a=9, b=-12, c=4

Discriminant =∆=b²-4ac

Discriminant =∆= (-12)²-4(9)(4)

Discriminant =∆= 144-144

Discriminant =∆=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:

 9x²-12x+4=0

9x²-6x-6x+4=0       ∴ 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 (b²-4ac<0)

If the discriminant is negative (b2-4ac<0) then√(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.x²+2x+3=0

Solution:

x²+2x+3=0 comparing with ax²+bx+c=0,we have

a=1 ,b=2,c=3

Discriminant =∆=b²-4ac

Discriminant =∆=(2)²-4(1)(3)

Discriminant =∆=4-12

Discriminant =∆=-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:

x²+2x+3=0

Comparing the standard form of quadratic equation

a=1,b=2,c=3

By using the quadratic formula

x=-b±√(b^2-4ac/2a

Put the value a, b and c

 x=-(2)±√(2^2-4(1)(3)/2(1)

x=-2±√(4-12/2

x=-2±√(-8/2

x=-2±√(4X2 i/2

x=-2±2√(2 i/2

x=2(-1±√(2 i)/2

x=-1±√2 i

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

Case:3. When discriminant is greater than zero or positive (b²-4ac>0)

If the discriminant is positive (b2-4ac>0) then √(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 2x²+5x-1=0

Solution:

2x²+5x-1=0 comparing with ax²+bx+c=0 we have

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

Discriminant =∆=b²-4ac

Discriminant =∆= (5)²-4(2)(-1)

 Discriminant =∆=25+8

Discriminant =∆=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 (b²-4ac>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) x²-px-q=0

Solution:

(p+q) x²-px-q=0 comparing with ax²+bx+c=0 we have

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

Discriminant =∆=b²-4ac

 Discriminant =∆=(-p)²-4(p+q)(-q)

 Discriminant =∆=p²+4pq+4q²

Discriminant =∆=p²+2(p)(2q)+(2q)²

Discriminant =∆= (p+q) ²

Which is a perfect square

Hence the roots of a given quadratic equation are

 Real, rational, and unequal

For example: x²+4x+3=0

Solution:

x²+4x+3=0 comparing with ax²+bx+c=0 we have

a=1 ,b=4 ,c=3

Discriminant =∆=b²-4ac

 Discriminant =∆=(4)²-4(1)(3)

Discriminant =∆=16-12

Discriminant =∆=4

Discriminant =∆=(2)² >0

Which is a perfect square

Solve the equation:

x²+4x+3=0

By using the factorization method

X²+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 (b²-4ac>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.4x²+6x+1=0

Solution:  

4x²+6x+1=0 comparing with ax²+bx+c=0 we have

a=4 ,b=6 ,c=1

Discriminant =∆=b²-4ac

Discriminant =∆= (6)²– 4(4)(1)

Discriminant =∆=36-16

Discriminant =∆= 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.