Chapter 9.7, Problem 39PFA

(a, b)

To determine

Solve the equation.

Answer to Problem 39PFA

  $\left(x,y\right)=\left(1,2\right),\left(3,4\right)$ the rocket and comet will collide at each other.

Explanation of Solution

Given information: $y=x^2-3x+4$ and $y-x=1$

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.

  $y=x^2-3x+4$ and $y-x=1$

So, simplify

  $y=x^2-3x+4$ and $y=x+1$

Now, equate

  $x^2-3x+4=x+1$

So, simplify

  $x^2-4x+3=0$

Now, factorize

  $x^2-x-3x+3=0$

So,

  $x\left(x-1\right)-3\left(x-1\right)=0$

Now,

  $\left(x-1\right)\left(x-3\right)=0$

Hence,

  $x=1$ or $x=3$

So, find value of $y$

  $y=x^2-3x+4$ or $y=x+1$

Now, put $x=1$ or $x=3$

  $y=1^2-\left(3\times 1\right)+4$ or $y=3+1$

Hence,

  $y=2$ or $y=4$

  $\left(x,y\right)=\left(1,2\right),\left(3,4\right)$ intersect at each other.

  

(c)

To determine

Solve the equation.

Answer to Problem 39PFA

  $\left(x,y\right)=\left(2,2\right)$ After change animator, the rocket and comet will collide at each other.

Explanation of Solution

Given information: $y=x^2-3x+4$ and $y=x$

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.

  $y=x^2-3x+4$ and $y=x$

Now, equate

  $x^2-3x+4=x$

So, simplify

  $x^2-4x+4=0$

Now, factorize

  ${\left(x-2\right)}^2=0$

Now,

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

Hence,

  $x=2$

So, find value of $y$

  $y=x^2-3x+4$ or $y=x$

Now, put $x=2$

  $y=2^2-\left(3\times 2\right)+4$ or $y=2$

Hence,

  $y=2$ or $y=4$

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