Chapter SH, Problem 9.25EP

To determine

To evaluate the value of discriminant and state the number of real solutions.

Answer to Problem 9.25EP

  $b^2-4ac=0$ .The equation has 1 real solution.

Explanation of Solution

Given information :

The function provided is $4x^2-12x=-9$

Formula used :

If the quadratic function $y=a{x}^2+bx+c$   has real solutions can be found out by using the quadratic formula:

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

The formula $b^2-4ac$ , also known as the discriminant can be used to determine the number of roots of solutions in a quadratic equation. There are three cases:

• $b^2-4ac<0$: The equation has 0 real solutions. The graph does not cross the  x -axis.
• $b^2-4ac=0$: The equation has 1 real solution. The graph touches the  x -axis at one point.
• $b^2-4ac>0$: The equation has 2 real solutions. The graph crosses through the  x -axis at two points.

Calculations :

  $4x^2-12x=-9$

The above expression can also be written as.

  $4x^2-12x+9=0$

Evaluating the discriminant.

  $b^2-4ac={(-12)}^2-4\times 4\times (9)$

Simplifying.

  $b^2-4ac=144-144$

Therefore, $b^2-4ac=0$ .The equation has 1 real solution.