Question

And also state the vertex and the axis of symmetry. I need help with number 2

Answer 1

In order to complete the square remember that

$(x\pm a)^2=x^2\pm2\cdot x\cdot a+a^2$

first, equal the expression divide all the expressions by 2,

$\begin{gathered} f(x)=2x^2+4x+7 \\ 2x^2+4x+7=0 \\ 2x^2+4x=-7 \\ x^2+2x=-\frac{7}{2} \end{gathered}$

then, we can find "a" using the second portion of the definition

$\begin{gathered} 2\cdot x\cdot a=2x \\ a=\frac{2x}{2x} \\ a=1 \end{gathered}$

then,

$a^2=1^2=1$

add 1 on both sides

$\begin{gathered} x^2+2x+1=-\frac{7}{2}+1 \\ x^2+2x+1=-\frac{5}{2} \end{gathered}$

rewrite as a square expression on the left,

$(x+1)^2=-\frac{5}{2}$

multiply both sides by 2

$2(x+1)^2=-5$

bring all to the left side and equal to f(x)

$\begin{gathered} f(x)=2(x+1)^2+5 \\ \text{The vertex is at (-1,5)} \\ \text{the axis of symmetry is x=-1} \end{gathered}$