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

And also state the vertex and the axis of symmetry. I need help with number 2-example-1

1 Answer

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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=-(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=(2x)/(2x) \\ a=1 \end{gathered}

then,


a^2=1^2=1

add 1 on both sides


\begin{gathered} x^2+2x+1=-(7)/(2)+1 \\ x^2+2x+1=-(5)/(2) \end{gathered}

rewrite as a square expression on the left,


(x+1)^2=-(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}

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