H Algebra 2/Trig · Unit 2
Find the missing number, write vertex form, and solve by completing the square.
Jump to section:
y = a(x − h)² + k
x − h = ±√k
Share the rectangles evenly on two sides of the square.
Complete the square.
9 units (pink) were needed to complete the square.
x² + 6x + 9 = (x + 3)(x + 3)
Use these steps for Complete the Square.
Inside the parentheses, move Right and Left opposite of what you think. So (x − 5)² moves Right 5.
Outside the parentheses, move Up and Down, is what it is. So +9 moves Up 9.
Vertex: (5, 9)
Set equation = 0.
x² + 6x − 8 = 0
Move number part over by adding opposites.
x² + 6x = 8
Complete the square.
x² + 6x + 9 = 8 + 9
(x + 3)² = 8 + 9
(x + 3)² = 17
Move number part back over.
y = (x + 3)² − 17
Vertex: (−3, −17)
Axis of symmetry: x = −3
Use these steps for Convert to Vertex Form, a = 1.
When a > 1, factor a from the x terms first, then use these same steps.
Use the missing-number steps. Find c, then factor.
Share the rectangles evenly on two sides of the square.
Half of 14 is 7.
The missing corner is 7 by 7.
49
49 units were needed to complete the square.
c = 49
Each side is x + 7.
Factor.
(x + 7)²
Answer: c = 49(x + 7)²
Share the rectangles evenly on two sides of the square.
Half of 10 is 5.
The missing corner is 5 by 5.
25
25 units were needed to complete the square.
c = 25
Each side is x + 5.
Factor.
(x + 5)²
Answer: c = 25(x + 5)²
Share the rectangles evenly on two sides of the square.
Half of −12 is −6.
The missing corner is 6 by 6.
36
36 units were needed to complete the square.
c = 36
Each side is x − 6.
Factor.
(x − 6)²
Answer: c = 36(x − 6)²
Share the rectangles evenly on two sides of the square.
Half of 8 is 4.
The missing corner is 4 by 4.
16
16 units were needed to complete the square.
c = 16
Each side is x + 4.
Factor.
(x + 4)²
Answer: c = 16(x + 4)²
Share the rectangles evenly on two sides of the square.
Half of −9.
(−9) · 12
−92
The missing corner is 92 by 92.
814
814 units were needed to complete the square.
c = 814
Each side is x − 92.
Factor.
(x − 92)²
Answer: c = 814(x − 92)²
Share the rectangles evenly on two sides of the square.
Half of 23.
23 · 12
13
The missing corner is 13 by 13.
19
19 units were needed to complete the square.
c = 19
Each side is x + 13.
Factor.
(x + 13)²
Answer: c = 19(x + 13)²
Share the rectangles evenly on two sides of the square.
Half of 43.
43 · 12
23
The missing corner is 23 by 23.
49
49 units were needed to complete the square.
c = 49
Each side is x + 23.
Factor.
(x + 23)²
Answer: c = 49(x + 23)²
Share the rectangles evenly on two sides of the square.
Half of −25.
(−25) · 12
−15
The missing corner is 15 by 15.
125
125 units were needed to complete the square.
c = 125
Each side is x − 15.
Factor.
(x − 15)²
Answer: c = 125(x − 15)²
Set the equation equal to 0, move the number, complete the square, then move it back. State the vertex and identify the axis of symmetry.
y = x² + 6x − 2
Set equation = 0.
x² + 6x − 2 = 0
Move number part over by adding opposites.
x² + 6x = 2
Complete the square.
x² + 6x + 9 = 2 + 9
(x + 3)² = 2 + 9
(x + 3)² = 11
Move number part back over.
y = (x + 3)² − 11
Vertex: (−3, −11)
Axis of symmetry: x = −3
Answer: y = (x + 3)² − 11Vertex: (−3, −11)Axis of symmetry: x = −3
y = x² − 8x + 3
Set equation = 0.
x² − 8x + 3 = 0
Move number part over by adding opposites.
x² − 8x = −3
Complete the square.
x² − 8x + 16 = −3 + 16
(x − 4)² = −3 + 16
(x − 4)² = 13
Move number part back over.
y = (x − 4)² − 13
Vertex: (4, −13)
Axis of symmetry: x = 4
Answer: y = (x − 4)² − 13Vertex: (4, −13)Axis of symmetry: x = 4
y = x² + 10x + 4
Set equation = 0.
x² + 10x + 4 = 0
Move number part over by adding opposites.
x² + 10x = −4
Complete the square.
x² + 10x + 25 = −4 + 25
(x + 5)² = −4 + 25
(x + 5)² = 21
Move number part back over.
y = (x + 5)² − 21
Vertex: (−5, −21)
Axis of symmetry: x = −5
Answer: y = (x + 5)² − 21Vertex: (−5, −21)Axis of symmetry: x = −5
y = x² − 4x − 7
Set equation = 0.
x² − 4x − 7 = 0
Move number part over by adding opposites.
x² − 4x = 7
Complete the square.
x² − 4x + 4 = 7 + 4
(x − 2)² = 7 + 4
(x − 2)² = 11
Move number part back over.
y = (x − 2)² − 11
Vertex: (2, −11)
Axis of symmetry: x = 2
Answer: y = (x − 2)² − 11Vertex: (2, −11)Axis of symmetry: x = 2
y = x² + 2x + 9
Set equation = 0.
x² + 2x + 9 = 0
Move number part over by adding opposites.
x² + 2x = −9
Complete the square.
x² + 2x + 1 = −9 + 1
(x + 1)² = −9 + 1
(x + 1)² = −8
Move number part back over.
y = (x + 1)² − (−8)
y = (x + 1)² + 8
Vertex: (−1, 8)
Axis of symmetry: x = −1
Answer: y = (x + 1)² + 8Vertex: (−1, 8)Axis of symmetry: x = −1
y = x² − 12x + 5
Set equation = 0.
x² − 12x + 5 = 0
Move number part over by adding opposites.
x² − 12x = −5
Complete the square.
x² − 12x + 36 = −5 + 36
(x − 6)² = −5 + 36
(x − 6)² = 31
Move number part back over.
y = (x − 6)² − 31
Vertex: (6, −31)
Axis of symmetry: x = 6
Answer: y = (x − 6)² − 31Vertex: (6, −31)Axis of symmetry: x = 6
Factor a from the x terms first, then use the same steps. State the vertex and identify the axis of symmetry.
y = 2x² + 12x − 5
Factor 2 from the x terms.
y = 2(x² + 6x) − 5
Set equation = 0.
2(x² + 6x) − 5 = 0
Move number part over by adding opposites.
2(x² + 6x) = 5
Complete the square.
Add 9 inside.
2 · 9 = 18
That goes on the right.
2(x² + 6x + 9) = 5 + 18
2(x + 3)² = 5 + 18
2(x + 3)² = 23
Move number part back over.
y = 2(x + 3)² − 23
Vertex: (−3, −23)
Axis of symmetry: x = −3
Answer: y = 2(x + 3)² − 23Vertex: (−3, −23)Axis of symmetry: x = −3
y = 3x² − 12x + 7
Factor 3 from the x terms.
y = 3(x² − 4x) + 7
Set equation = 0.
3(x² − 4x) + 7 = 0
Move number part over by adding opposites.
3(x² − 4x) = −7
Complete the square.
Add 4 inside.
3 · 4 = 12
That goes on the right.
3(x² − 4x + 4) = −7 + 12
3(x − 2)² = −7 + 12
3(x − 2)² = 5
Move number part back over.
y = 3(x − 2)² − 5
Vertex: (2, −5)
Axis of symmetry: x = 2
Answer: y = 3(x − 2)² − 5Vertex: (2, −5)Axis of symmetry: x = 2
y = 2x² − 8x + 3
Factor 2 from the x terms.
y = 2(x² − 4x) + 3
Set equation = 0.
2(x² − 4x) + 3 = 0
Move number part over by adding opposites.
2(x² − 4x) = −3
Complete the square.
Add 4 inside.
2 · 4 = 8
That goes on the right.
2(x² − 4x + 4) = −3 + 8
2(x − 2)² = −3 + 8
2(x − 2)² = 5
Move number part back over.
y = 2(x − 2)² − 5
Vertex: (2, −5)
Axis of symmetry: x = 2
Answer: y = 2(x − 2)² − 5Vertex: (2, −5)Axis of symmetry: x = 2
y = 4x² + 16x − 1
Factor 4 from the x terms.
y = 4(x² + 4x) − 1
Set equation = 0.
4(x² + 4x) − 1 = 0
Move number part over by adding opposites.
4(x² + 4x) = 1
Complete the square.
Add 4 inside.
4 · 4 = 16
That goes on the right.
4(x² + 4x + 4) = 1 + 16
4(x + 2)² = 1 + 16
4(x + 2)² = 17
Move number part back over.
y = 4(x + 2)² − 17
Vertex: (−2, −17)
Axis of symmetry: x = −2
Answer: y = 4(x + 2)² − 17Vertex: (−2, −17)Axis of symmetry: x = −2
y = 2x² + 6x − 4
Factor 2 from the x terms.
y = 2(x² + 3x) − 4
Set equation = 0.
2(x² + 3x) − 4 = 0
Move number part over by adding opposites.
2(x² + 3x) = 4
Complete the square.
Add 94 inside.
2 · 94 = 184
92
That goes on the right.
2(x² + 3x + 94) = 4 + 92
2(x + 32)² = 4 + 92
2(x + 32)² = 82 + 92
2(x + 32)² = 172
Move number part back over.
y = 2(x + 32)² − 172
Vertex: (−32, −172)
Axis of symmetry: x = −32
Answer: y = 2(x + 32)² − 172Vertex: (−32, −172)Axis of symmetry: x = −32
y = 3x² + 6x + 2
Factor 3 from the x terms.
y = 3(x² + 2x) + 2
Set equation = 0.
3(x² + 2x) + 2 = 0
Move number part over by adding opposites.
3(x² + 2x) = −2
Complete the square.
Add 1 inside.
3 · 1 = 3
That goes on the right.
3(x² + 2x + 1) = −2 + 3
3(x + 1)² = −2 + 3
3(x + 1)² = 1
Move number part back over.
y = 3(x + 1)² − 1
Vertex: (−1, −1)
Axis of symmetry: x = −1
Answer: y = 3(x + 1)² − 1Vertex: (−1, −1)Axis of symmetry: x = −1
Two with a = 1, two with a > 1, then two whose vertex is a fraction.
x² + 4x − 12 = 0
Move number part over by adding opposites.
x² + 4x = 12
Complete the square.
x² + 4x + 4 = 12 + 4
(x + 2)² = 12 + 4
(x + 2)² = 16
The formula is
x − h = ±√k
Substitute.
x + 2 = ±√16
x + 2 = ±4
x = −2 + 4
x = 2
x = −2 − 4
x = −6
Answer: x = 2 and x = −6
x² + 8x − 9 = 0
Move number part over by adding opposites.
x² + 8x = 9
Complete the square.
x² + 8x + 16 = 9 + 16
(x + 4)² = 9 + 16
(x + 4)² = 25
The formula is
x − h = ±√k
Substitute.
x + 4 = ±√25
x + 4 = ±5
x = −4 + 5
x = 1
x = −4 − 5
x = −9
Answer: x = 1 and x = −9
x² − 10x + 16 = 0
Move number part over by adding opposites.
x² − 10x = −16
Complete the square.
x² − 10x + 25 = −16 + 25
(x − 5)² = −16 + 25
(x − 5)² = 9
The formula is
x − h = ±√k
Substitute.
x − 5 = ±√9
x − 5 = ±3
x = 5 + 3
x = 8
x = 5 − 3
x = 2
Answer: x = 8 and x = 2
2x² − 12x + 10 = 0
Divide by 2.
x² − 6x + 5 = 0
Move number part over by adding opposites.
x² − 6x = −5
Complete the square.
x² − 6x + 9 = −5 + 9
(x − 3)² = −5 + 9
(x − 3)² = 4
The formula is
x − h = ±√k
Substitute.
x − 3 = ±√4
x − 3 = ±2
x = 3 + 2
x = 5
x = 3 − 2
x = 1
Answer: x = 5 and x = 1
3x² − 6x − 24 = 0
Divide by 3.
x² − 2x − 8 = 0
Move number part over by adding opposites.
x² − 2x = 8
Complete the square.
x² − 2x + 1 = 8 + 1
(x − 1)² = 8 + 1
(x − 1)² = 9
The formula is
x − h = ±√k
Substitute.
x − 1 = ±√9
x − 1 = ±3
x = 1 + 3
x = 4
x = 1 − 3
x = −2
Answer: x = 4 and x = −2
x² − 3x − 4 = 0
Move number part over by adding opposites.
x² − 3x = 4
Complete the square.
x² − 3x + 94 = 4 + 94
(x − 32)² = 4 + 94
(x − 32)² = 164 + 94
(x − 32)² = 254
The formula is
x − h = ±√k
Substitute.
x − 32 = ±√(254)
x − 32 = ±52
x = 32 + 52
x = 82
x = 4
x = 32 − 52
x = −22
x = −1
Answer: x = 4 and x = −1
2x² − 10x + 8 = 0
Divide by 2.
x² − 5x + 4 = 0
Move number part over by adding opposites.
x² − 5x = −4
Complete the square.
x² − 5x + 254 = −4 + 254
(x − 52)² = −4 + 254
(x − 52)² = −164 + 254
(x − 52)² = 94
The formula is
x − h = ±√k
Substitute.
x − 52 = ±√(94)
x − 52 = ±32
x = 52 + 32
x = 82
x = 4
x = 52 − 32
x = 22
x = 1
Answer: x = 4 and x = 1
One of each kind, mixed together. If an answer is wrong, the solution names what to practice.
2x² + 4x − 16 = 0
Divide by 2.
x² + 2x − 8 = 0
Move number part over by adding opposites.
x² + 2x = 8
Complete the square.
x² + 2x + 1 = 8 + 1
(x + 1)² = 8 + 1
(x + 1)² = 9
The formula is
x − h = ±√k
Substitute.
x + 1 = ±√9
x + 1 = ±3
x = −1 + 3
x = 2
x = −1 − 3
x = −4
If this was wrong, practice Solve by Completing the Square, a > 1.
Answer: x = 2 and x = −4
Share the rectangles evenly on two sides of the square.
Half of −18 is −9.
The missing corner is 9 by 9.
81
81 units were needed to complete the square.
c = 81
Each side is x − 9.
Factor.
(x − 9)²
If this was wrong, practice Complete the Square.
Answer: c = 81(x − 9)²
y = x² − 6x − 4
Set equation = 0.
x² − 6x − 4 = 0
Move number part over by adding opposites.
x² − 6x = 4
Complete the square.
x² − 6x + 9 = 4 + 9
(x − 3)² = 4 + 9
(x − 3)² = 13
Move number part back over.
y = (x − 3)² − 13
Vertex: (3, −13)
Axis of symmetry: x = 3
If this was wrong, practice Convert to Vertex Form, a = 1.
Answer: y = (x − 3)² − 13Vertex: (3, −13)Axis of symmetry: x = 3
x² + 7x + 6 = 0
Move number part over by adding opposites.
x² + 7x = −6
Complete the square.
x² + 7x + 494 = −6 + 494
(x + 72)² = −6 + 494
(x + 72)² = −244 + 494
(x + 72)² = 254
The formula is
x − h = ±√k
Substitute.
x + 72 = ±√(254)
x + 72 = ±52
x = −72 + 52
x = −22
x = −1
x = −72 − 52
x = −122
x = −6
If this was wrong, practice Solve by Completing the Square, fraction vertex, a = 1.
Answer: x = −1 and x = −6
y = 3x² + 12x − 1
Factor 3 from the x terms.
y = 3(x² + 4x) − 1
Set equation = 0.
3(x² + 4x) − 1 = 0
Move number part over by adding opposites.
3(x² + 4x) = 1
Complete the square.
Add 4 inside.
3 · 4 = 12
That goes on the right.
3(x² + 4x + 4) = 1 + 12
3(x + 2)² = 1 + 12
3(x + 2)² = 13
Move number part back over.
y = 3(x + 2)² − 13
Vertex: (−2, −13)
Axis of symmetry: x = −2
If this was wrong, practice Convert to Vertex Form, a > 1.
Answer: y = 3(x + 2)² − 13Vertex: (−2, −13)Axis of symmetry: x = −2
x² − 2x − 15 = 0
Move number part over by adding opposites.
x² − 2x = 15
Complete the square.
x² − 2x + 1 = 15 + 1
(x − 1)² = 15 + 1
(x − 1)² = 16
The formula is
x − h = ±√k
Substitute.
x − 1 = ±√16
x − 1 = ±4
x = 1 + 4
x = 5
x = 1 − 4
x = −3
If this was wrong, practice Solve by Completing the Square, a = 1.
Answer: x = 5 and x = −3
2x² + 2x − 12 = 0
Divide by 2.
x² + x − 6 = 0
Move number part over by adding opposites.
x² + x = 6
Complete the square.
x² + x + 14 = 6 + 14
(x + 12)² = 6 + 14
(x + 12)² = 244 + 14
(x + 12)² = 254
The formula is
x − h = ±√k
Substitute.
x + 12 = ±√(254)
x + 12 = ±52
x = −12 + 52
x = 42
x = 2
x = −12 − 52
x = −62
x = −3
If this was wrong, practice Solve by Completing the Square, fraction vertex, a > 1.
Answer: x = 2 and x = −3
Need help completing the square?
Book a Session with Amy