Answer:
The x-intercepts (there are 2 of them) are located at (3±√2, 0)
Step-by-step explanation:
In order to find the x-intercepts, we have to factor the equation to solve it for x. However, at the present time, we have no equation to factor; we only have the vertex (h, k) and a coordinate (0, 7). So we will use those to find the equation of the parabola. If you graph the points, it's apparent that this is a positive x-squared parabola of the vertex form:
[tex]y=a(x-h)^2+k[/tex]
We need to solve for a to get the correct equation. Filling in our info gives us:
[tex]7=a(0-3)^2-2[/tex] so
7 = a(9) - 2 and
9 = 9a so
a = 1. The equation for our parabola is
[tex]y=(x-3)^2-2[/tex]
The easiest way to find the x-intercepts (factor it) is to write it in standard form which is
[tex]y=ax^2+bx+c[/tex]
In order to do that we have to expand that binomial by FOILing and we get
[tex]y=x^2-6x+9-2[/tex] which simplifies to
[tex]y=x^2-6x+7[/tex]
In order to factor that you have to throw it into the quadratic formula. That looks like this:
[tex]x=\frac{6+/-\sqrt{-6^2-4(1)(7)} }{2(1)}[/tex]
which simplifies to
[tex]x=\frac{6+/-\sqrt{8} }{2}[/tex]
The square root of 8 simplifies:
[tex]x=\frac{6+/-2\sqrt{2} }{2}[/tex]
and dividing everything but the radicand (the number under the square root) by 2 gives you both of your x-intercepts:
x = 3 ± √2
