Find the x-intercept of the parabola with vertex (3,-2) and y-intercept (0,7)

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] so7 = a(9) - 2 and9 = 9a soa = 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