Question

How do you solve the equation `x^2+8=-6x` by graphing?

Answer 1

See below.

Explanation:

To graph `x^2+8=-6x`

First add `-6x` to both sides to get:

`x^2+6x+8=0`

We can now write this in function form by replacing the `0` with a `y`:

`x^2 +6x+8= y` or. `y=x^2 +6x+8`

Find any intercepts ( not the x axis intercepts, since we are going to find them from the graph):

`y` axis intercepts occur where `x = 0`, so:

`0^2+6(0) + 8 = 8`

So `y` axis intercept at `(0 , 8 )`

We can now find the vertex. This will give us the coordinates of the highest or lowest point on the parabola.

To find the vertex we need to get `y=x^2 +6x+8` into the form `y = a(x-color(blue)(h))^2 + color(red)(k)`.

Where `a` is the coefficient of `x^2` `color(blue)(h)` is the axis of symmetry and `color(red)(k)` is the minimum/maximum value.

To find the vertex:
Bracket off the terms containing the variable:

`(x^2 + 6x ) + 8`

Factor out the coefficient of `x^2` if this is not `1`
Add the square of half the coefficient of `x` inside the bracket and subtract it outside the bracket:

`(x^2 +6x +(3)^2) - (3)^2+8`

Change `(x^2 +6x +(3)^2)` into the square of a binomial:

`(x^2 + 3)^2-(3)^2+8`

Collect terms outside the bracket:

`(x^2 + 3)^2-1`

This shows that `color(blue)(h)= -3` and `color(blue)(k)=-1`

So the coordinate of the vertex is `( -3 , -1 )`

To find other points we will have to put in some values for `x` and calculate the corresponding values of `y`

Examples:

`x = 1=>y= 15`
`x = 2=>y= 24`
`x = -1=>y= 3`
`x = -5=>y = 4`
`x=-7=>y= 15`

All our coordinates including the vertex and `y` intercept are:

`( -3 , -1 )` `( 1 , 15 )` `( 4 , 24 )` `( -1 , 3 )` `( -5 , 3 )` `(-7 , 15 )` `( 0 , 8 )`

Plot these points and draw the parabola:

Join points:

Roots will be where parabola crosses the `x` axis. This looks to be at `(-4 , 0 )` and `( -2 , 0 )`