How do you write #g(x)=2x^2+8x+13# in vertex form?

Question

How do you write #g(x)=2x^2+8x+13# in  vertex form?

Savannah Anderson · Accepted Answer

#g(x)=2(x+2)^2+5# #"given the parabola in standard form "ax^2+bx+c# #"the x-coordinate of the vertex is"# #x_(color(red)"vertex")=-b/(2a)# #2x^2+8x+13" is in standard form"# #"with "a=2,b=8,c=13# #rArrx_(color(red)"vertex")=-8/4=-2# #"substitute this value into the equation for y-coordinate"# #rArry_(color(red)"vertex")=2(-2)^2+8(-2)+13=5# #rArrcolor(magenta)"vertex "=(-2,5)# #"the equation of the parabola in "color(blue)"vertex form"# is. #•color(white)(x)y=a(x-h)^2+k# #"where "(h,k)" are the coordinates of the vertex and a"# #"is a constant"# #"here "a=2" and "(h,k)=(-2,5)# #rArrg(x)=2(x+2)^2+5larrcolor(red)" in vertex form"#

Jace Anderson · Answer

undefined To write the quadratic function $ g(x) = 2x^2 + 8x + 13 $ in vertex form, complete the square. First, factor out the leading coefficient from the $ x^2 $ and $ x $ terms. Then, complete the square by adding and subtracting the square of half the coefficient of the $ x $ term. Finally, rewrite the expression in vertex form by grouping the squared term with its related terms.

Here's the process:

1. Factor out the leading coefficient from the $ x^2 $ and $ x $ terms:
$$ g(x) = 2(x^2 + 4x) + 13 $$

2. Complete the square within the parentheses:
$$ g(x) = 2(x^2 + 4x + 4 - 4) + 13 $$

3. Rewrite the expression by grouping the squared term with its related terms:
$$ g(x) = 2(x^2 + 4x + 4) - 8 + 13 $$

4. Factor the perfect square trinomial:
$$ g(x) = 2(x + 2)^2 + 5 $$

So, the quadratic function $ g(x) = 2x^2 + 8x + 13 $ in vertex form is $ g(x) = 2(x + 2)^2 + 5 $.