The quadratic function given to us is:
[tex]g(x)=2x^2+20x+49[/tex]
We are asked to find the vertex form of the function.
The general formula for the vertex form of a quadratic equation is:
[tex]\begin{gathered} g(x)=a(x-h)^2+k \\ \text{where,} \\ (h,k)\text{ is the coordinate of the vertex} \end{gathered}[/tex]
In order to write the function in its vertex form, we need to perform a couple of operations on the function.
1. Add and subtract the square of the half of the coefficient of x to the function.
2. Factor out the function with its repeated roots and re-write the equation.
Now, let us solve.
1. Add and subtract the square of the half of the coefficient of x to the function.
[tex]\begin{gathered} g(x)=2x^2+20x+49=2(x^2+10x+\frac{49}{2}) \\ \text{half of coefficient of x:} \\ \frac{10}{2}=5 \\ \text{square of the half of the coefficient of x:} \\ 5^2=25 \\ \\ \therefore g(x)=2(x^2+10x+25-25+\frac{49}{2}) \end{gathered}[/tex]
2. Factor out the function with its repeated roots and re-write the equation.
[tex]\begin{gathered} g(x)=2(x^2+10x+25)-2(25+\frac{49}{2}) \\ re-\text{write the above function} \\ g(x)=2(x^2+10x+25)-1 \\ \text{Let us factorize this:} \\ g(x)=2(x+5)^2-1 \end{gathered}[/tex]
Therefore, we can conclude that the Equation and vertex of the equation is:
[tex]\begin{gathered} Equation\colon g(x)=2(x+5)^2-1 \\ \\ Vertex\colon(-5,-1) \end{gathered}[/tex]
