Chapter 3, Problem 7CT

To determine

To determine: The explanation to find the number and type of solutions of the quadratic equation. Also, use the discriminant to justify the answer.

Answer to Problem 7CT

The discriminant of the quadratic equation is greater than zero and equation has two solutions.

Explanation of Solution

Given information:

The quadratic equation: $-x^2+\frac{1}{2}x+\frac{3}{2}=0$

The graph of the equation:

  

Formula used:

Discriminant formula:

The discriminant D of a quadratic equation $a{x}^2+bx+c=0$ is,

  $D=b^2-4ac$

Discriminant conditions:

If $D<0$ ; then the roots of the quadratic equations are not real.

If $D=0$ ; then the roots of the quadratic equation are real and equal.

If $D>0$ ; then the roots of the quadratic equations are distinct real numbers.

Calculation:

The given quadratic equation is $-x^2+\frac{1}{2}x+\frac{3}{2}=0$ .

It is seen in the graph that the curve of the quadratic equation intersects the x -axis at two points. This means the equation has two solutions.

Also, the discriminant of the quadratic equation is greater than zero.

To find the discriminant of the given equation, use the discriminant formula $D=b^2-4ac$ .

Substitute −1 for a , $\frac{1}{2}$ for b and $\frac{3}{2}$ for c in the above formula and simplify.

  $\begin{array}{c}D={\left(\frac{1}{2}\right)}^2{-}^2\cancel{4}\left(-1\right)\left(\frac{3}{\cancel{2}}\right)\\ =\frac{1}{4}+2\left(3\right)\\ =\frac{1}{4}+6\\ =\frac{25}{4}\end{array}$

It is seen that the discriminant is greater than zero.

So, the equation has two solutions.

Hence, the discriminant of the quadratic equation is greater than zero and equation has two solutions.