Chapter 9.7, Problem 33HP

(a)

To determine

Solve the equation.

Answer to Problem 33HP

  $\left(x,y\right)=\left(2,1\right)$ intersect at each other.

Explanation of Solution

Given information: $y=x^2-2x+1$ and $y=2x-3$

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.

  $y=x^2-2x+1$ and $y=2x-3$

Now, equate

  $x^2-2x+1=2x-3$

So, simplify

  $x^2-4x+4=0$

Now, factorize

  $x^2-2x-2x+4=0$

So,

  $x\left(x-2\right)-2\left(x-2\right)=0$

Now,

  $\left(x-2\right)\left(x-2\right)=0$

Hence,

  $x=2$ or $x=2$

So, find value of $y$

  $y=x^2-2x+1$ or $y=2x-3$

Now, put $x=2$ or $x=2$

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

Hence,

  $y=1$ or $y=1$

  $\left(x,y\right)=\left(2,1\right)$ intersect at each other.

  

(b)

To determine

Solve the equation.

Answer to Problem 33HP

Not intersect at each other.

Explanation of Solution

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

Formula Used: Graphing Utility, Factorization and $D=b^2-4ac$

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.

  $y=x^2-2x+5$ and $y=2x$

Now, equate

  $x^2-2x+5=2x$

So, simplify

  $x^2-4x+5=0$

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

As $a=1,b=-4$ and $c=5$

  $D={\left(-4\right)}^2-4\times 1\times 5$

Now,

  $D=16-20$

Hence,

  $D=-4$

As, $D$ equal to negative so $x$ has no real solution.

  

(c)

To determine

Solve the equation.

Answer to Problem 33HP

  $\left(x,y\right)=\left(4.83,14.67\right),\left(-0.83,3.34\right)$ intersect at each other.

Explanation of Solution

Given information: $y=x^2-2x+1$ and $y=2x+5$

Formula Used: Graphing Utility, $D=b^2-4ac$ and $x=\frac{-b\pm \sqrt{D}}{2a}$

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.

  $y=x^2-2x+1$ and $y=2x+5$

Now, equate

  $x^2-2x+1=2x+5$

So, simplify

  $x^2-4x-4=0$

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

As $a=1,b=-4$ and $c=-4$

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

Now,

  $D=16+16$

Hence,

  $D=32$

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

  $x=\frac{-b\pm \sqrt{D}}{2a}$

So,

  $x=\frac{-\left(-4\right)\pm \sqrt{32}}{2\times 1}$

Now, square root of 32 is 5.66.

  $x=\frac{4\pm 5.66}{2}$

Now,

  $x=\frac{4+5.66}{2}$ or $x=\frac{4-5.66}{2}$

Now,

  $x=\frac{9.66}{2}$ or $x=\frac{-1.66}{2}$

Now,

  $x=4.83$ or $x=-0.83$

So, find value of $y$

  $y=x^2-2x+1$ or $y=2x+5$

Now, put $x=4.83$ or $x=-0.83$

  $y={\left(4.83\right)}^2-\left(2\times 4.83\right)+1$ or $y=\left(2\times \left(-0.83\right)\right)+5$

Hence,

  $y=14.67$ or $y=3.34$

  $\left(x,y\right)=\left(4.83,14.67\right),\left(-0.83,3.34\right)$ intersect at each other.