Answer:
[tex]f(x)=2(x+9)^2-5[/tex]Explanation:
The vertex form of a quadratic equation is given as:
[tex]y=a(x-h)^2+k[/tex]Given the equation
[tex]f\mleft(x\mright)=2x^2+36x+157[/tex]First, we rewrite it as follows:
[tex]f\mleft(x\mright)-157=2x^2+36x[/tex]Next, we factorize the right-hand side.
[tex]f(x)-157=2(x^2_{}+18x)[/tex]We complete the expression in the bracket at the right-hand side.
[tex]f(x)-157+2(81)=2(x^2+18x+81)[/tex]This then gives us:
[tex]\begin{gathered} f(x)-157+162=2(x+9)^2 \\ f(x)+5=2(x+9)^2 \\ f(x)=2(x+9)^2-5 \end{gathered}[/tex]The vertex form is:
[tex]f(x)=2(x+9)^2-5[/tex]