How do you write the vertex form equation of the parabola #(x+5)(x+4)#?

Question

Savannah Anderson · Accepted Answer

#y=(x-(-9/2))^2+(-1/4)#

The general vertex form for a parabola (with a vertical axis of symmetry) is #color(white)("XXX")y=color(green)(m)(x-color(red)(a))^2+color(blue)(b)# with vertex at #(color(red)(a),color(blue)(b))# (#color(green)(m)# can be thought of as a parameter that effects the "spread" of the parabola).

Given: #color(white)("XXX")y=(x+5)(x+4)#

This can be rewritten as: #color(white)("XXX")y=x^2+9x+20#

Completing the square: #color(white)("XXX")y=x^2+9xcolor(brown)(+(9/2)^2)+20color(brown)(-(9/2)^2)#

#color(white)("XXX")y=(x+9/2)^2+(80/4-81/4)#

#color(white)("XXX")y=color(green)(1)(x-color(red)(""(-9/2)))^2+color(blue)(""(-1/4))# with vertex at #(color(red)(-9/2),color(blue)(-1/4))#

(Note that when #color(green)(m)=1# it is often omitted.)

graph{(x+5)(x+4) [-5.95, -0.474, -1.327, 1.41]}

Jameson Allen · Answer

undefined To write the vertex form equation of the parabola given by the factors (x+5)(x+4), you first expand the expression to get the standard form equation of the parabola. Then, you complete the square to convert it to vertex form.

First, expand the expression:
(x+5)(x+4) = x^2 + 4x + 5x + 20 = x^2 + 9x + 20

Now, complete the square:
x^2 + 9x + 20 = (x + (9/2))^2 - (81/4) + 20

So, the vertex form equation of the parabola is:
y = (x + (9/2))^2 - (81/4) + 20