Completing the Square Day 2
Convert the equation from standard to vertex form.
f(x) = 3x2 + 18x + 7
(3x2 + 18x + __) + 7 - __
Write two blanks: one +, one -
3(x2 + 6x + __) + 7 – 3__
Factor the GCF. The GCF will
mult. the outside blank.
3(x2
2
+ 6x + 3 ) + 7 - 3 3
2
Fill the blanks with
3(x2 + 6x + 9) + 7 – 3(9)
Square the parenthesis
3(x2 + 6x + 9) + 7 – 27
Multiply
3(x + 3)2 – 20
Factor and simplify
𝑏 2
2
Convert the equation from standard to vertex form.
g(x) = -5x2 + 50x + 128
(-5x2 + 50x + __) + 128 - __Write two blanks: one +, one -
-5(x2 – 10x + __) + 128 – -5__Factor the GCF. The GCF will
mult. the outside blank.
-5(x2
2
- 10x + −5 ) + 128 – -5 −5
2
Fill the blanks with
𝑏 2
2
-5(x2 - 10x + 25) + 128 – -5(25)
Square the parenthesis
-5(x2 - 10x + 25) + 128 + 125
Multiply
-5(x - 5)2 + 253
Factor and simplify
Convert the equation from standard to vertex form.
h(x) = 2x2 + 9x – 13
(2x2 + 9x + __) – 13 - __
Write two blanks: one +, one -
we
2(x2 + 4.5x + __) – 13 - 2__ Factor out a 2 because
2
want it to be 1x .
2(x2
2(x2
2
+ 4.5x + 2.25 ) – 13 - 2 2.25
2
Fill the blanks with
+ 4.5x + 5.0625) – 13 - 2(5.0625)
2(x2 + 4.5x + 5.0625) – 13 - 10.125
2(x + 2.25)2 – 23.125
Square the
parenthesis
Multiply
Factor and simplify
𝑏 2
2
Convert the equation from standard to vertex form.
f(x) = 80x2 – 64x + 1
(80x2 – 64x + __) + 1 - __ Write two blanks: one +, one -
80(x2 – 0.8x + __) + 1 - 80__ Factor out a 80 2because we
want it to be 1x .
80(x2
80(x2
2
– 0.8x + −.4 ) + 1 - 80 −.4
– 0.8x + .16) + 1 - 80(.16)
80(x2 – 0.8x + .16) + 1 – 12.8
y = 80(x – .4)2 – 11.8
2
Fill the blanks with
Square the
parenthesis
Multiply
Factor and simplify
𝑏 2
2
An alternative method.
In standard form, the x-coordinate of the vertex
−𝑏
for a parabola can be found using x = .
2𝑎
Convert f(x) = 80x2 – 64x + 1 to vertex form.
x=
x=
−𝑏
2𝑎
−(−64)
2(80)
Find the x-coordinate of the vertex.
=
64
160
= 0.4
y = 80(.4)2 – 64(.4) + 1
y = 11.8
Find the y-coordinate of the vertex,
by plugging in the x-coordinate into
the original equation.
Vertex: (0.4, 11.8)
a = 80
Now we know the vertex.
y = a(x – h)2 + k
y = 80(x – 0.4)2 + 11.8
Plug-in a, h, and k into vertex form.
a = 80 comes from the original
equation.
Convert f(x) = 2x2 +9x – 13 to vertex form.
x=
x=
y=
−𝑏
2𝑎
−(9)
2(2)
Find the x-coordinate of the vertex.
=
−9
4
2(-2.25)2
= -2.25
+ 9(-2.25) – 13
y = -23.125
Find the y-coordinate of the
vertex, by plugging in the xcoordinate into the original
equation.
Vertex: (-2.25, -23.125)
a=2
Now we know the vertex.
y = a(x – h)2 + k
y = 2(x + 2.25)2 – 23.125
Plug-in a, h, and k into vertex form.
a = 80 comes from the original
equation.