Chapter 9.5, Problem 59E

a.

To determine

To find the value of c for which the equation has two solutions.

Answer to Problem 59E

The value of c should be greater than 9.

Explanation of Solution

Given:

Equation:

  $4x^2+12x+c=0$

Calculation:

On comparing the equation with $a{x}^2+bx+c$

  $a=4,b=12$

Finding the value of c.

For equation to have two solution discriminant must be greater than 0

Therefore,

  $b^2-4ac$ >0

Substituting the values,

  ${(12)}^2-4(4)c>0$

Opening brackets,

  $144-16c>0$

So,

  $-16c>-144$

Therefore,

  $c>\frac{-144}{-16}$

So,

  $c>9$

Therefore, the value of c should be greater than 9.

b.

To determine

To find the value of c for which the equation has one solutions.

Answer to Problem 59E

The value of c should be equal to 9.

Explanation of Solution

Given:

Equation:

  $4x^2+12x+c=0$

Calculation:

On comparing the equation with $a{x}^2+bx+c$

  $a=4,b=12$

Finding the value of c.

For equation to have two solution discriminant must be equal to 0

Therefore,

  $b^2-4ac$ =0

Substituting the values,

  ${(12)}^2-4(4)c=0$

Opening brackets,

  $144-16c=0$

So,

  $-16c=-144$

Therefore,

  $c=\frac{-144}{-16}$

So,

  $c=9$

Therefore, the value of c should be equal to 9.

c.

To determine

To find the value of c for which the equation has no solutions.

Answer to Problem 59E

The value of c should be less than 9.

Explanation of Solution

Given:

Equation:

  $4x^2+12x+c=0$

Calculation:

On comparing the equation with $a{x}^2+bx+c$

  $a=4,b=12$

Finding the value of c.

For equation to have two solution discriminant must be less than 0

Therefore,

  $b^2-4ac$<0

Substituting the values,

  ${(12)}^2-4(4)c<0$

Opening brackets,

  $144-16c<0$

So,

  $-16c<-144$

Therefore,

  $c<\frac{-144}{-16}$

So,

  $c<9$

Therefore, the value of c should be less than 9.