Question

Function: y = x² + 5x - 7Vertex:Solutions:) and

Answer 1

For a quadratic function of the form:

$\begin{gathered} y=ax^2+bx+c \\ the_{\text{ }}vertex: \\ V=(h,k) \\ can_{\text{ }}be_{\text{ }}found_{\text{ }}as_{\text{ }}follows: \\ h=-\frac{b}{2a} \\ k=y(h) \end{gathered}$

So, for:

$\begin{gathered} y=x^2+5x-7 \\ a=1 \\ b=5 \\ c=-7 \end{gathered}$$\begin{gathered} h=-\frac{5}{2(1)}=-\frac{5}{2}=-2.5 \\ k=y(2.5)=(-2.5^2)+5(-2.5)-7=-13.25 \end{gathered}$

Therefore, the vertex is:

V = (-2.5,-13.25)

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The solutions can be found using the quadratic formula:

$\begin{gathered} x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\ \\ x=\frac{-5\pm\sqrt{5^2-4(1)(-7)}}{2(1)} \\ \\ x=\frac{-5\pm\sqrt{53}}{2} \end{gathered}$

Therefore, the solutions are:

$\begin{gathered} x=\frac{-5+\sqrt{53}}{2}\approx1.14 \\ and \\ x=\frac{-5-\sqrt{53}}{2}\approx-6.14 \end{gathered}$