Chapter 9.7, Problem 38PFA

To determine

Solve the equations.

Answer to Problem 38PFA

Option D

  $\left(x,y\right)=\left(0.4,-0.56\right),\left(-2.4,-3.4\right)$

Explanation of Solution

Given Information: $y=x-1$ and $y=-x^2-x$

Formula Used: Factorization

Calculation:

  $y=x-1$ and $y=-x^2-x$

Now, equate

  $x-1=-x^2-x$

So, simplify

  $x^2+2x-1=0$

Now, use formula $D=b^2-4ac$ to check if solution of $x$ exist or not.

As $a=1,b=2$ and $c=-1$

  $D={\left(2\right)}^2-4\times 1\times \left(-1\right)$

So,

  $D=4+4$

Hence,

  $D=8$

As, $D$ equal to positive so $x$ has real roots.

  $x=\frac{-b\pm \sqrt{D}}{2a}$

So,

  $x=\frac{-\left(2\right)\pm \sqrt{8}}{2\times 1}$

Now, square root of 8 is 2.83.

  $x=\frac{-2\pm 2.83}{2}$

Now,

  $x=\frac{-2+2.83}{2}$ or $x=\frac{-2-2.83}{2}$

Now,

  $x=\frac{0.83}{2}$ or $x=\frac{-4.83}{2}$

Now,

  $x=0.4$ or $x=-2.4$

So, find value of $y$

  $y=-x^2-x$ or $y=x-1$

Now, put $x=0.4$ or $x=-2.4$

  $y=-{\left(0.4\right)}^2-0.4$ or $y=-2.4-1$

Hence,

  $y=-0.56$ or $y=-3.4$

So, $\left(x,y\right)=\left(0.4,-0.56\right),\left(-2.4,-3.4\right)$