Chapter SH, Problem 9.27EP

To determine

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

Answer to Problem 9.27EP

Discriminant is 1. $b^2-4ac>0$ .The equation has 2 real solutions

Explanation of Solution

Given information :

The functions provided is $y=x^2+2x+1$ and $y-x=1$

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 :

  $y=x^2+2x+1$

Substituting the value of $y=x+1$ .

  $x+1=x^2+2x+1$

Simplifying.

  $x^2+x=0$

Evaluating the discriminant.

  $b^2-4ac={(1)}^2-4\times 1\times (0)$

Simplifying.

  $b^2-4ac=1$

Therefore,

  $b^2-4ac>0$ .The equation has 2 real solutions