Chapter 4, Problem 19PT

(a)

To determine

To find: The value of discriminant.

Answer to Problem 19PT

The value of discriminant is $49$ .

Explanation of Solution

Given information:

The given equation is $6x^2+7x=0$ .

Calculation:

Consider the equation $6x^2+7x=0$ .

The value of discriminant $b^2-4ac$ is calculated below.

  $\begin{array}{c}{b}^2-4ac=7^2-4\left(6\right)\left(0\right)\\ =49>0\end{array}$

Therefore, the value of discriminant is $49$ .

(b)

To determine

To find: The number type of roots.

Answer to Problem 19PT

The two roots are real and rational number .

Explanation of Solution

Given information:

  $6x^2+7x=0$

Calculation:

Calculate the number of type roots.

The discriminant obtained in part $a$ .is a perfect square $\left(49=7^2\right)$ and thus the two roots are real and rational number.

Therefore, the two roots are real and rational number.

(c)

To determine

To find: The exact solution of quadratic equation.

Answer to Problem 19PT

The exact value of quadratic equation is $x=0$ and $x=-\frac{7}{6}$ .

Explanation of Solution

Given information:

The given information is $6x^2+7x=0$ .

Calculation:

The exact solution is found below by using $a=6,b=7,c=0$ and the quadratic equation. $\begin{array}{c}x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ =\frac{-7\pm \sqrt{{\left(7\right)}^2-4\left(6\right)\left(0\right)}}{2\left(6\right)}\\ =\frac{-7+7}{12}or\frac{-7-7}{12}\\ =0or-\frac{7}{6}\end{array}$

The roots are $x=0$ and $x=-\frac{7}{6}$ .

Therefore the exact value of quadratic equation is $x=0$ and $x=-\frac{7}{6}$ .