Question

How do you use the graph to solve `0=9x^2+12x-7`?

Answer 1

You have to graph the function `y = 9x^2 + 12x - 7` and then locate the x-intercepts. These will be the solutions to the equation.

To graph, start by completing the square.

`y = 9x^2 + 12x - 7`

`y = 9(x^2 + 4/3 + n - n) - 7`

`n = (b/2)^2 = ((4/3)/2)^2 = 16/36 = 4/9`

`y = 9(x^2 + 4/3 + 4/9 - 4/9) - 7`

`y = 9(x^2 + 4/3 + 4/9) - 4 - 7`

`y = 9(x + 2/3)^2 - 11`

Graph this parabola, using the vertex, intercepts, the domain and range, the axis of symmetry and the direction of opening.

Once this is done, you should end up with the following parabola.

Looking at the x-intercepts, you will find the solutions are irrational. They are `x ~= 0.4` and `x ~=-1.8`

I would strongly recommend finding the solutions algebraically after, since it's such an easy way to verify your answer. Just take the `√`, since we're in vertex form.

`0 = 9(x + 2/3)^2 - 11`

`11/9 = (x + 2/3)^2`

`+-sqrt(11/9) = x + 2/3`

`x = -2/3 +- sqrt(11)/3`

`x = (-2 +- sqrt(11))/3`

The decimal approximation for these solutions are:

`x = 0.43887` and `x = -1.77221`

So, our approximations were quite close to the actual solutions. However, they were not nearly as precise as `x = (2 +- sqrt(11))/3`, so perhaps when the solutions aren't rational your teacher won't make you solve the equations graphically on evaluations.

Hopefully this helps!