Question

How do you find the x intercepts of `36x^2 + 84x + 49 = 0`?

Answer 1

`y=36x^2+84x+49` has a single `x`-intercept at `x=-7/6`

Explanation:

Technically an equation in one variable does not have intercepts; it has a (or multiple) solution(s).

`36x^2+84x+49`
can be factored as `(6x+7)^2 or (6x+7)*(6x+7)`

So if `36x^2+84x+49=0`
then `(6x+7)*(6x+7)=0`
which implies `(6x+7)=0`
`color(white)("xxxxxxxx")x=-7/6`

Answer 2

See a solution process below:

Explanation:

We can factor the left side of the equation as:

`(6x + 7)^2 = 0`

Or

`(6x + 7)(6x + 7) = 0`

Because both terms on the left are the same there will be only one `x` intercept. We can solve one of the terms for `0`:

`6x + 7 = 0`

`6x + 7 - color(red)(7) = 0 - color(red)(7)`

`6x + 0 = -7`

`6x = -7`

`(6x)/color(red)(6) = -7/color(red)(6)`

`(color(red)(cancel(color(black)(6)))x)/cancel(color(red)(6)) = -7/color(red)(6)`

`x = -7/6`

As you can see from the graph the parabola touches the x-axis at just one point: `-7/6` or `(-7/6, 0)`

graph{(y - 36x^2 - 84x - 49)((x+7/6)^2+(y)^2-0.0005) = 0 [-2, 0, -0.5, 0.51]}