Question

How do you solve `x^2 + 9x = -7` graphically and algebraically?

Answer 1

`x approx -0.86 or -8.14`

Explanation:

`x^2+9x =-7`

`x^2+9x+7=0`

Algebraic Solution:

This is a quadratic eqation of the form: `ax^2+bx+c=0`

The quadratic formula states:

`x=(-b+-sqrt(b^2-4ac))/(2a)`

Hence, `x= (-9+-sqrt(9^2-4*1*7))/(2*1)`

`=(-9+-sqrt(81-28))/2`

`= (-9+-sqrt(53))/2`

`approx (-9+-7.28)/2`

`approx -4.5+-3.64`

Hence, `x approx -0.86 or -8.14`

Graphical Solution:

To find the zeros of `f(x) = x^2+9x+7` we plot `f(x)` and find the `x-`intercepts. As below:

graph{x^2+9x+7 [-28.87, 28.85, -14.44, 14.43]}

As can be seen, the `x-`intercepts of `f(x)` are `approx -0.86 and -8.14`

Hence, this is the graphical solution to the given equation.

Answer 2

`x=(-9 +- sqrt(53))/2`

Explanation:

`x^2+9x=-7`
`x^2 + 9x + 7 = 0`
`x=(-b +- sqrt(b^2 - 4ac))/(2a)`
`x=(-(9) +- sqrt((9)^2 - 4(1)(7)))/(2(1))`
`x=(-9 +- sqrt(53))/2`

Solving the equation graphically would be inefficient, since the algebraic method is required to use the graphical method. However, if you are given a graph with the exact value of the x-intercepts, the x-intercepts will be the solutions to the equation.