Chapter 9.7, Problem 29HP

To determine

Solve the equation.

Answer to Problem 29HP

  $\left(x,y\right)=\left(1,-1\right),\left(-\frac{5}{2},-\frac{25}{4}\right)$ intersect at each other.

Explanation of Solution

Given information: $y=x^2+3x-5$ and $y=-x^2$

Formula Used: Graphing Utility and Factorization

Calculation:

For such equations, a graphing utility can often be used to investigate possible solutions. When a graphing utility is used to solve an equation, usually approximate solutions are obtained.

  $y=x^2+3x-5$ and $y=-x^2$

Now, equatex²

  $x^2+3x-5=-x^2$

So, simplify

  $2x^2+3x-5=0$

Now, factorize

  $2x^2-2x+5x-5=0$

So,

  $2x\left(x-1\right)+5\left(x-1\right)=0$

Now,

  $\left(x-1\right)\left(2x+5\right)=0$

Hence,

  $x=1$ or $x=-\frac{5}{2}$

So, find value of $y$

  $y=x^2+3x-5$ or $y=-x^2$

Now, put $x=1$ or $x=-\frac{5}{2}$

  $y=1^2+\left(3\times 1\right)-5$ or $y=-{\left(-\frac{5}{2}\right)}^2$

Hence,

  $y=-1$ or $y=-\frac{25}{4}$

  $\left(x,y\right)=\left(1,-1\right),\left(-\frac{5}{2},-\frac{25}{4}\right)$ intersect at each other.