Chapter 4, Problem 52SGR

(a)

To determine

To find: The value of discriminant.

Answer to Problem 52SGR

The value of discriminant is $D=144$ .

Explanation of Solution

Given information:

The given equation is $x^2+4x-32=0$ .

Calculation:

Consider the equation $x^2+4x-32=0$ .

Compare above expression with standard form $a{x}^2+bx+c$ .

Where,

  $a=1$ , $b=4$ , $c=-32$ .

Consider following expression for discriminant.

  $D=b^2-4ac$ …..(1)

Substitute the values in the equation(1)

  $\begin{array}{c}D=b^2-4ac\\ ={\left(4\right)}^2-4\left(1\right)\left(-32\right)\\ =16+128\\ =144\end{array}$

Therefore, the value of discriminant is $D=144$ .

(b)

To determine

To find: The number type of roots.

Answer to Problem 52SGR

The number of roots is $2$ which is real and rational.

Explanation of Solution

Given information:

The given equation is $x^2+4x-32=0$

Calculation:

Calculate the number of type roots.

The expression is $x=\frac{-b\pm \sqrt{D}}{2a}$ .

Substitute the value of $D$ in the expression.

  $\begin{array}{c}x=\frac{-b\pm \sqrt{144}}{2a}\\ =\frac{-b\pm 12}{2a}\end{array}$

Therefore, the number of roots is $2$ which is real and rational.

(c)

To determine

To find: The exact solution of quadratic formula.

Answer to Problem 52SGR

The exact value of quadratic equation is $x=7.5,3.5$

Explanation of Solution

Given information:

The given equation is $x^2+4x-32=0$ .

Calculation:

Substitute the value of $D$

  $\begin{array}{c}x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ =\frac{-4\pm 12}{2\left(1\right)}\\ =7.5,3.5\end{array}$

Therefore the exact value of quadratic equation is $x=7.5,3.5$ .