Chapter P.5, Problem 57E

Solution

a.

The number and nature of the zeros of quadratic polynomial $a{x}^2+bx+c$ , if $b^2-4ac>0$ .

Two distinct and real zeros.

Given:

The discriminant is $b^2-4ac>0$ .

Concept used:

• The discriminant is given by, $D=b^2-4ac$
• For a quadratic equation $a{x}^2+bx+c=0$ the number of roots depend on:
• If discriminant is negative, $b^2-4ac<0$, then there is no real roots
• If discriminant is zero, $b^2-4ac=0$, then there are two same real roots
• If discriminant is positive, $b^2-4ac>0$, then there are two distinct real roots

Calculation:

If the discriminant of the quadratic polynomial is positive, then there are two real and different zeros for the given quadratic polynomial.

b.

The number and nature of the zeros of quadratic polynomial $a{x}^2+bx+c$ , if $b^2-4ac=0$ .

Two same real zeros.

Given:

The discriminant is $b^2-4ac=0$ .

Concept used:

• The discriminant is given by, $D=b^2-4ac$
• For a quadratic equation $a{x}^2+bx+c=0$ the number of roots depend on:
• If discriminant is negative, $b^2-4ac<0$, then there is no real roots
• If discriminant is zero, $b^2-4ac=0$, then there are two same real roots
• If discriminant is positive, $b^2-4ac>0$, then there are two distinct real roots

Calculation:

If the discriminant of the quadratic polynomial is equal to zero, then there are two same real zeros for the given quadratic polynomial.

c.

The number and nature of the zeros of quadratic polynomial $a{x}^2+bx+c$ , if $b^2-4ac<0$ .

No real zeros. Two complex zeros.

Given:

The discriminant is $b^2-4ac<0$ .

Concept used:

• The discriminant is given by, $D=b^2-4ac$
• For a quadratic equation $a{x}^2+bx+c=0$ the number of roots depend on:
• If discriminant is negative, $b^2-4ac<0$, then there are no real roots
• If discriminant is zero, $b^2-4ac=0$, then there are two same real roots
• If discriminant is positive, $b^2-4ac>0$, then there are two distinct real roots

Calculation:

If the discriminant of the quadratic polynomial is negative, then there are no real zeros in that case. When the discriminant is negative, the quadratic polynomial has two imaginary zeros.