Question

Using a graphing utility, find the exact solutions of the system. Round to the nearest hundredth and choose a solution to the system from the choices below.

y = x² + 3x
y = 2x + 5

(0,4)
(2.79, 0.58)
(1.79, 8.58)
(-0.58, -2.79)

Answer 1

Part 1) The exact solutions are

`(\frac{-1+√(21)} {2},4+√(21))` and
`(\frac{-1-√(21)} {2},4-√(21))`

Part 2) (1.79, 8.58)

Explanation:

we have

`y=x^(2) +3x` ----> equation A

`y=2x+5` ----> equation B

we know that

When solving the system of equations by graphing, the solution of the system is the intersection points both graphs

Find the exact solutions of the system

equate equation A and equation B

`x^(2) +3x=2x+5\\x^(2) +3x-2x-5=0\\x^(2) +x-5=0`

The formula to solve a quadratic equation of the form

`ax^(2) +bx+c=0`

is equal to

`x=\frac{-b\pm\sqrt{b^(2)-4ac}} {2a}`

in this problem we have

`x^(2) +x-5=0`

so

`a=1\\b=1\\c=-5`

substitute in the formula

`x=\frac{-1\pm\sqrt{1^(2)-4(1)(-5)}} {2(1)}`

`x=\frac{-1\pm√(21)} {2}`

so

The solutions are

`x_1=\frac{-1+√(21)} {2}`

`x_2=\frac{-1-√(21)} {2}`

Find the values of y

First solution

For
`x_1=\frac{-1+√(21)} {2}`

`y=2(\frac{-1+√(21)} {2})+5`

`y=-1+√(21)+5\\\\y=4+√(21)`

The first solution is the point
`(\frac{-1+√(21)} {2},4+√(21))`

Second solution

For
`x_2=\frac{-1-√(21)} {2}`

`y=2(\frac{-1-√(21)} {2})+5`

`y=-1-√(21)+5\\\\y=4-√(21)`

The second solution is the point
`(\frac{-1-√(21)} {2},4-√(21))`

Round to the nearest hundredth

First solution

`(\frac{-1+√(21)} {2},4+√(21))` ----->
`(1.79,8.58)`

`(\frac{-1-√(21)} {2},4-√(21))` ----->
`(-2.79,-0.58)`

see the attached figure to better understand the problem