Download Notes

Completing the Square
Standard form of a quadratic:
𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
a is the stretch/compression, reflection
c is the y-intercept
Convert y = 2(x – 1)2 + 4 into standard form
y = 2(x – 1)(x – 1) + 4
FOIL
y = 2(x2 – 2x + 1) + 4
Distribute
y = 2x2 – 4x + 2 + 4
Combine like terms
y = 2x2 – 4x + 6
Convert y =
y=
y=
y=
y=
1
− (x
2
– 4)2 – 3 into standard form.
1
− (x – 4)(x – 4) – 3
2
1 2
− (x – 8x + 16) – 3
2
1 2
− x
2
1 2
− x
2
FOIL
+ 4x – 8 – 3
Distribute
+ 4x – 11
Combine like terms
Completing the square is a factoring technique
used to write a quadratic in vertex form.
Consider: (x + 3)2
x2 + 6x + 9
(x – 6)2
x2 – 12x + 36
Factor: x2 – 8x + 16
x2 + 18x + 81
(x – 4)2
(x + 9)2
To complete the square, the number inside the
parenthesis is half of the middle term.
x2
+ 10x + 25 → (x +
10 2
)
2
→ (x + 5)2
What if the numbers don’t match?
x2 + 10x + 32
32 should be a 25
(x2 + 10x + __) + 32 - __
(x2 + 10x +
10 2
)
2
+ 32 -
(x2 + 10x + 25) + 32 - 25
(x + 5)2 + 7
Write two blanks: one +, one 10 2
𝑏 2
Fill the blanks with
2
2
Square the parenthesis
Factor and simplify
Complete the square:
70 should be 64
x2 – 16x + 70
(x2 – 16x + __) + 70 - __
(x2 – 16x +
16 2
)
2
+ 70 -
(x2 – 16x + 64) + 70 - 64
(x – 8)2 + 6
Write two blanks: one +, one 16 2
𝑏 2
Fill the blanks with
2
2
Square the parenthesis
Factor and simplify
Complete the square:
111 should be 121
x2 + 22x + 111
(x2 + 22x + __) + 111 - __ Write two blanks: one +, one (x2 + 22x +
22 2
)
2
+ 111 -
22 2
𝑏 2
Fill the blanks with
2
2
(x2 + 22x + 121) + 111 - 121 Square the parenthesis
(x + 11)2 – 10
Factor and simplify
Write the equation of the parabola in vertex form.
y = x2 + 3x – 3
(x2 + 3x + __) – 3 - __
(x2
+ 3x +
3 2
)
2
–3-
3 2
2
Write two blanks: one +, one Fill the blanks with
(x2 + 3x + 2.25) – 3 – 2.25 Square the parenthesis
(x + 1.5)2 – 5.25
Factor and simplify
𝑏 2
2
Write the equation of the parabola in vertex form.
y = x2 – 9x + 1
(x2 – 9x + __) + 1 - __
(x2
– 9x +
−9 2
)
2
+1-
Write two blanks: one +, one −9 2
2
(x2 – 9x + 20.25) + 1 – 20.25
(x – 4.5)2 – 5.25
Fill the blanks with
𝑏 2
2
Square the parenthesis
Factor and simplify