H Algebra 2/Trig · Unit 2
Write vertex form when a is greater than 1, solve when the square needs a fraction, and complete the square when x² is negative.
Jump to section:
f(x) = a(x − h)² + k
x − h = ±
Factor a from the x terms.
Set equation = 0.
Move the number part over by adding opposites.
Complete the square inside the parentheses.
Multiply that corner by a. That product goes on the right.
Move the number part back over.
Divide by a.
Take half of the new x term, then square that half.
Add the square on both sides.
The formula is
x − h = ±
Move the x² term over by adding opposites.
Complete the square with the positive x².
Factor a from the x terms, then complete the square. The first three add an integer. The last three add a fraction.
f(x) = 5x² − 20x + 3
Factor 5 from the x terms.
f(x) = 5(x² − 4x) + 3
Set equation = 0.
5(x² − 4x) + 3 = 0
Move number part over by adding opposites.
5(x² − 4x) = −3
Complete the square.
Add 4 inside.
5 · 4 = 20
That goes on the right.
5(x² − 4x + 4) = −3 + 20
5(x − 2)² = −3 + 20
5(x − 2)² = 17
Move number part back over.
f(x) = 5(x − 2)² − 17
Vertex: (2, −17)
Answer: f(x) = 5(x − 2)² − 17Vertex: (2, −17)
f(x) = 2x² + 8x + 5
Factor 2 from the x terms.
f(x) = 2(x² + 4x) + 5
Set equation = 0.
2(x² + 4x) + 5 = 0
Move number part over by adding opposites.
2(x² + 4x) = −5
Complete the square.
Add 4 inside.
2 · 4 = 8
That goes on the right.
2(x² + 4x + 4) = −5 + 8
2(x + 2)² = −5 + 8
2(x + 2)² = 3
Move number part back over.
f(x) = 2(x + 2)² − 3
Vertex: (−2, −3)
Answer: f(x) = 2(x + 2)² − 3Vertex: (−2, −3)
f(x) = 3x² − 24x + 1
Factor 3 from the x terms.
f(x) = 3(x² − 8x) + 1
Set equation = 0.
3(x² − 8x) + 1 = 0
Move number part over by adding opposites.
3(x² − 8x) = −1
Complete the square.
Add 16 inside.
3 · 16 = 48
That goes on the right.
3(x² − 8x + 16) = −1 + 48
3(x − 4)² = −1 + 48
3(x − 4)² = 47
Move number part back over.
f(x) = 3(x − 4)² − 47
Vertex: (4, −47)
Answer: f(x) = 3(x − 4)² − 47Vertex: (4, −47)
f(x) = 4x² + 24x − 7
Factor 4 from the x terms.
f(x) = 4(x² + 6x) − 7
Set equation = 0.
4(x² + 6x) − 7 = 0
Move number part over by adding opposites.
4(x² + 6x) = 7
Complete the square.
Add 9 inside.
4 · 9 = 36
That goes on the right.
4(x² + 6x + 9) = 7 + 36
4(x + 3)² = 7 + 36
4(x + 3)² = 43
Move number part back over.
f(x) = 4(x + 3)² − 43
Vertex: (−3, −43)
Answer: f(x) = 4(x + 3)² − 43Vertex: (−3, −43)
f(x) = 2x² + 6x + 1
Factor 2 from the x terms.
f(x) = 2(x² + 3x) + 1
Set equation = 0.
2(x² + 3x) + 1 = 0
Move number part over by adding opposites.
2(x² + 3x) = −1
Complete the square.
Half of 3.
3 · 12
32
The missing corner is 32 by 32.
94
Add 94 inside.
2 · 94 = 184
= 92
That goes on the right.
2(x² + 3x + 94) = −1 + 92
2(x + 32)² = −1 + 92
2(x + 32)² = −22 + 92
2(x + 32)² = 72
Move number part back over.
f(x) = 2(x + 32)² − 72
Vertex: (−32, −72)
Answer: f(x) = 2(x + 32)² − 72Vertex: (−32, −72)
f(x) = 3x² − 4x + 2
Factor 3 from the x terms.
f(x) = 3(x² − 43x) + 2
Set equation = 0.
3(x² − 43x) + 2 = 0
Move number part over by adding opposites.
3(x² − 43x) = −2
Complete the square.
Half of −43.
(−43) · 12
−23
The missing corner is 23 by 23.
49
Add 49 inside.
3 · 49 = 129
= 43
That goes on the right.
3(x² − 43x + 49) = −2 + 43
3(x − 23)² = −2 + 43
3(x − 23)² = −63 + 43
3(x − 23)² = −23
Move number part back over.
f(x) = 3(x − 23)² − (−23)
f(x) = 3(x − 23)² + 23
Vertex: (23, 23)
Answer: f(x) = 3(x − 23)² + 23Vertex: (23, 23)
f(x) = 4x² + 4x − 3
Factor 4 from the x terms.
f(x) = 4(x² + x) − 3
Set equation = 0.
4(x² + x) − 3 = 0
Move number part over by adding opposites.
4(x² + x) = 3
Complete the square.
Half of 1.
1 · 12
12
The missing corner is 12 by 12.
14
Add 14 inside.
4 · 14 = 44
= 1
That goes on the right.
4(x² + x + 14) = 3 + 1
4(x + 12)² = 3 + 1
4(x + 12)² = 4
Move number part back over.
f(x) = 4(x + 12)² − 4
Vertex: (−12, −4)
Answer: f(x) = 4(x + 12)² − 4Vertex: (−12, −4)
Divide by a, then complete the square. The number you add is a fraction.
4x² − 3x = 10
Divide by 4.
x² − 34x = 104
x² − 34x = 52
Complete the square.
Half of −34.
(−34) · 12
−38
The missing corner is 38 by 38.
964
x² − 34x + 964 = 52 + 964
(x − 38)² = 52 + 964
(x − 38)² = 16064 + 964
(x − 38)² = 16964
The formula is
x − h = ±
Substitute.
x − 38 = ±
x − 38 = ±138
x = 38 + 138
x = 168
x = 2
x = 38 − 138
x = −108
x = −54
Answer: x = 2 and x = −54
2x² + 3x = 5
Divide by 2.
x² + 32x = 52
Complete the square.
Half of 32.
32 · 12
34
The missing corner is 34 by 34.
916
x² + 32x + 916 = 52 + 916
(x + 34)² = 52 + 916
(x + 34)² = 4016 + 916
(x + 34)² = 4916
The formula is
x − h = ±
Substitute.
x + 34 = ±
x + 34 = ±74
x = −34 + 74
x = 44
x = 1
x = −34 − 74
x = −104
x = −52
Answer: x = 1 and x = −52
3x² − 5x = 2
Divide by 3.
x² − 53x = 23
Complete the square.
Half of −53.
(−53) · 12
−56
The missing corner is 56 by 56.
2536
x² − 53x + 2536 = 23 + 2536
(x − 56)² = 23 + 2536
(x − 56)² = 2436 + 2536
(x − 56)² = 4936
The formula is
x − h = ±
Substitute.
x − 56 = ±
x − 56 = ±76
x = 56 + 76
x = 126
x = 2
x = 56 − 76
x = −26
x = −13
Answer: x = 2 and x = −13
4x² + 5x = 6
Divide by 4.
x² + 54x = 64
x² + 54x = 32
Complete the square.
Half of 54.
54 · 12
58
The missing corner is 58 by 58.
2564
x² + 54x + 2564 = 32 + 2564
(x + 58)² = 32 + 2564
(x + 58)² = 9664 + 2564
(x + 58)² = 12164
The formula is
x − h = ±
Substitute.
x + 58 = ±
x + 58 = ±118
x = −58 + 118
x = 68
x = 34
x = −58 − 118
x = −168
x = −2
Answer: x = 34 and x = −2
5x² − 2x = 3
Divide by 5.
x² − 25x = 35
Complete the square.
Half of −25.
(−25) · 12
−15
The missing corner is 15 by 15.
125
x² − 25x + 125 = 35 + 125
(x − 15)² = 35 + 125
(x − 15)² = 1525 + 125
(x − 15)² = 1625
The formula is
x − h = ±
Substitute.
x − 15 = ±
x − 15 = ±45
x = 15 + 45
x = 55
x = 1
x = 15 − 45
x = −35
Answer: x = 1 and x = −35
2x² + x = 6
Divide by 2.
x² + 12x = 62
x² + 12x = 3
Complete the square.
Half of 12.
12 · 12
14
The missing corner is 14 by 14.
116
x² + 12x + 116 = 3 + 116
(x + 14)² = 3 + 116
(x + 14)² = 4816 + 116
(x + 14)² = 4916
The formula is
x − h = ±
Substitute.
x + 14 = ±
x + 14 = ±74
x = −14 + 74
x = 64
x = 32
x = −14 − 74
x = −84
x = −2
Answer: x = 32 and x = −2
The x² term is a negative integer. Move it over so you complete the square with a positive x².
−2x² = 4x + 7
Move the x² term over by adding opposites.
Add 2x² on both sides.
0 = 2x² + 4x + 7
Write the positive x² on the left.
2x² + 4x + 7 = 0
Divide by 2.
x² + 42x + 72 = 0
x² + 2x + 72 = 0
Move number part over by adding opposites.
x² + 2x = −72
Complete the square.
x² + 2x + 1 = −72 + 1
(x + 1)² = −72 + 1
(x + 1)² = −72 + 22
(x + 1)² = −52
The formula is
x − h = ±
Substitute.
x + 1 = ±
k is negative, so the answers use i.
x + 1 = ± i
Simplify the square root.
=
Rationalize the denominator.
· ·
2
x + 1 = ±2 i
x = −1 + 2 i
x = −1 − 2 i
Answer: x = −1 + 2 i and x = −1 − 2 i
−2x² = 8x + 6
Move the x² term over by adding opposites.
Add 2x² on both sides.
0 = 2x² + 8x + 6
Write the positive x² on the left.
2x² + 8x + 6 = 0
Divide by 2.
x² + 4x + 3 = 0
Move number part over by adding opposites.
x² + 4x = −3
Complete the square.
x² + 4x + 4 = −3 + 4
(x + 2)² = −3 + 4
(x + 2)² = 1
The formula is
x − h = ±
Substitute.
x + 2 = ±
x + 2 = ±1
x = −2 + 1
x = −1
x = −2 − 1
x = −3
Answer: x = −1 and x = −3
−3x² = 9x − 12
Move the x² term over by adding opposites.
Add 3x² on both sides.
0 = 3x² + 9x − 12
Write the positive x² on the left.
3x² + 9x − 12 = 0
Divide by 3.
x² + 3x − 4 = 0
Move number part over by adding opposites.
x² + 3x = 4
Complete the square.
Half of 3.
3 · 12
32
The missing corner is 32 by 32.
94
x² + 3x + 94 = 4 + 94
(x + 32)² = 4 + 94
(x + 32)² = 164 + 94
(x + 32)² = 254
The formula is
x − h = ±
Substitute.
x + 32 = ±
x + 32 = ±52
x = −32 + 52
x = 22
x = 1
x = −32 − 52
x = −82
x = −4
Answer: x = 1 and x = −4
−4x² = −12x + 8
Move the x² term over by adding opposites.
Add 4x² on both sides.
0 = 4x² − 12x + 8
Write the positive x² on the left.
4x² − 12x + 8 = 0
Divide by 4.
x² − 3x + 2 = 0
Move number part over by adding opposites.
x² − 3x = −2
Complete the square.
Half of −3.
(−3) · 12
−32
The missing corner is 32 by 32.
94
x² − 3x + 94 = −2 + 94
(x − 32)² = −2 + 94
(x − 32)² = −84 + 94
(x − 32)² = 14
The formula is
x − h = ±
Substitute.
x − 32 = ±
x − 32 = ±12
x = 32 + 12
x = 42
x = 2
x = 32 − 12
x = 22
x = 1
Answer: x = 2 and x = 1
−5x² = 15x + 10
Move the x² term over by adding opposites.
Add 5x² on both sides.
0 = 5x² + 15x + 10
Write the positive x² on the left.
5x² + 15x + 10 = 0
Divide by 5.
x² + 3x + 2 = 0
Move number part over by adding opposites.
x² + 3x = −2
Complete the square.
Half of 3.
3 · 12
32
The missing corner is 32 by 32.
94
x² + 3x + 94 = −2 + 94
(x + 32)² = −2 + 94
(x + 32)² = −84 + 94
(x + 32)² = 14
The formula is
x − h = ±
Substitute.
x + 32 = ±
x + 32 = ±12
x = −32 + 12
x = −22
x = −1
x = −32 − 12
x = −42
x = −2
Answer: x = −1 and x = −2
−6x² = 30x + 36
Move the x² term over by adding opposites.
Add 6x² on both sides.
0 = 6x² + 30x + 36
Write the positive x² on the left.
6x² + 30x + 36 = 0
Divide by 6.
x² + 5x + 6 = 0
Move number part over by adding opposites.
x² + 5x = −6
Complete the square.
Half of 5.
5 · 12
52
The missing corner is 52 by 52.
254
x² + 5x + 254 = −6 + 254
(x + 52)² = −6 + 254
(x + 52)² = −244 + 254
(x + 52)² = 14
The formula is
x − h = ±
Substitute.
x + 52 = ±
x + 52 = ±12
x = −52 + 12
x = −42
x = −2
x = −52 − 12
x = −62
x = −3
Answer: x = −2 and x = −3
Need help with these completing the square problems?
Book a Session with Amy