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Amy Ferguson Moncure

H Algebra 2/Trig · Unit 2

Quadratic Formula

Solve ax² + bx + c = 0 with the quadratic formula. Leave answers in simplest radical form, including i.

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Key Ideas Easy Medium A Medium B Hard Quiz

x = −b ± √(b² − 4ac)2a

Quadratic Formula
x = −b ± √(b² − 4ac)2a

x² + 2x − 3 = 0

a = 1
b = 2
c = −3

x = −2 ± √(2² − 4(1)(−3))2

x = −2 ± √(4 − (−12))2

x = −2 ± √(4 + 12)2

x = −2 ± √162

x = −2 ± 42

x = −2 + 42

x = 22

x = 1

x = −2 − 42

x = −62

x = −3

x = 1 and x = −3

Careful!!
Careful when b is negative!!2x² − 3x − 1 = 0
a = 2
b = −3
c = −1

x = −b ± √(b² − 4ac)2a

x = −(−3) ± √((−3)² − 4(2)(−1))4

x = 3 ± √(9 − (−8))4

x = 3 ± √(9 + 8)4

x = 3 ± √174

x = 3 ± √174

Ready???
Is the Equation Ready???Set equation equal to zero first.x² = −4x + 1

x² = −4x + 1

x² + 4x − 1 = 0

a = 1
b = 4
c = −1

x = −b ± √(b² − 4ac)2a

x = −4 ± √(4² − 4(1)(−1))2

x = −4 ± √(16 − (−4))2

x = −4 ± √(16 + 4)2

x = −4 ± √202

x = −4 ± 2√52

x = −2 ± √5

x = −2 ± √5

Level 1

Easy

The equation is set equal to zero. Integer or fraction solutions.

1
Solve by the quadratic formula. x² + 4x + 3 = 0
Show Answer
a = 1
b = 4
c = 3

x = −b ± √(b² − 4ac)2a

x = −4 ± √(4² − 4(1)(3))2

x = −4 ± √(16 − 12)2

x = −4 ± √42

x = −4 ± 22

x = −4 + 22

x = −22

x = −1

x = −4 − 22

x = −62

x = −3

Answer: x = −1 and x = −3

2
Solve by the quadratic formula. 2x² − 3x − 2 = 0
Show Answer
a = 2
b = −3
c = −2

x = −b ± √(b² − 4ac)2a

x = −(−3) ± √((−3)² − 4(2)(−2))4

x = 3 ± √(9 − (−16))4

x = 3 ± √(9 + 16)4

x = 3 ± √254

x = 3 ± 54

x = 3 + 54

x = 84

x = 2

x = 3 − 54

x = −24

x = −12

Answer: x = 2 and x = −12

3
Solve by the quadratic formula. x² − 5x − 6 = 0
Show Answer
a = 1
b = −5
c = −6

x = −b ± √(b² − 4ac)2a

x = −(−5) ± √((−5)² − 4(1)(−6))2

x = 5 ± √(25 − (−24))2

x = 5 ± √(25 + 24)2

x = 5 ± √492

x = 5 ± 72

x = 5 + 72

x = 122

x = 6

x = 5 − 72

x = −22

x = −1

Answer: x = 6 and x = −1

4
Solve by the quadratic formula. 3x² + 4x − 4 = 0
Show Answer
a = 3
b = 4
c = −4

x = −b ± √(b² − 4ac)2a

x = −4 ± √(4² − 4(3)(−4))6

x = −4 ± √(16 − (−48))6

x = −4 ± √(16 + 48)6

x = −4 ± √646

x = −4 ± 86

x = −4 + 86

x = 46

x = 23

x = −4 − 86

x = −126

x = −2

Answer: x = 23 and x = −2

5
Solve by the quadratic formula. x² − 10x + 25 = 0
Show Answer
a = 1
b = −10
c = 25

x = −b ± √(b² − 4ac)2a

x = −(−10) ± √((−10)² − 4(1)(25))2

x = 10 ± √(100 − 100)2

x = 10 ± √02

x = 10 ± 02

x = 102

x = 5

Answer: x = 5

Level 2A

Medium A

The equation is set equal to zero. Two solutions with radicals.

1
Solve by the quadratic formula. x² + 4x − 1 = 0
Show Answer
a = 1
b = 4
c = −1

x = −b ± √(b² − 4ac)2a

x = −4 ± √(4² − 4(1)(−1))2

x = −4 ± √(16 − (−4))2

x = −4 ± √(16 + 4)2

x = −4 ± √202

x = −4 ± 2√52

x = −2 ± √5

Answer: x = −2 ± √5

2
Solve by the quadratic formula. x² − 6x + 4 = 0
Show Answer
a = 1
b = −6
c = 4

x = −b ± √(b² − 4ac)2a

x = −(−6) ± √((−6)² − 4(1)(4))2

x = 6 ± √(36 − 16)2

x = 6 ± √202

x = 6 ± 2√52

x = 3 ± √5

Answer: x = 3 ± √5

3
Solve by the quadratic formula. 2x² + 2x − 1 = 0
Show Answer
a = 2
b = 2
c = −1

x = −b ± √(b² − 4ac)2a

x = −2 ± √(2² − 4(2)(−1))4

x = −2 ± √(4 − (−8))4

x = −2 ± √(4 + 8)4

x = −2 ± √124

x = −2 ± 2√34

x = −1 ± √32

Answer: x = −1 ± √32

4
Solve by the quadratic formula. 3x² − 4x − 1 = 0
Show Answer
a = 3
b = −4
c = −1

x = −b ± √(b² − 4ac)2a

x = −(−4) ± √((−4)² − 4(3)(−1))6

x = 4 ± √(16 − (−12))6

x = 4 ± √(16 + 12)6

x = 4 ± √286

x = 4 ± 2√76

x = 2 ± √73

Answer: x = 2 ± √73

5
Solve by the quadratic formula. x² + 2x − 5 = 0
Show Answer
a = 1
b = 2
c = −5

x = −b ± √(b² − 4ac)2a

x = −2 ± √(2² − 4(1)(−5))2

x = −2 ± √(4 − (−20))2

x = −2 ± √(4 + 20)2

x = −2 ± √242

x = −2 ± 2√62

x = −1 ± √6

Answer: x = −1 ± √6

Level 2B

Medium B

The equation is set equal to zero. Imaginary solutions.

1
Solve by the quadratic formula. x² + 2x + 5 = 0
Show Answer
a = 1
b = 2
c = 5

x = −b ± √(b² − 4ac)2a

x = −2 ± √(2² − 4(1)(5))2

x = −2 ± √(4 − 20)2

x = −2 ± √(−16)2

x = −2 ± i√162

x = −2 ± 4i2

x = −1 ± 2i

Answer: x = −1 ± 2i

2
Solve by the quadratic formula. x² − 6x + 13 = 0
Show Answer
a = 1
b = −6
c = 13

x = −b ± √(b² − 4ac)2a

x = −(−6) ± √((−6)² − 4(1)(13))2

x = 6 ± √(36 − 52)2

x = 6 ± √(−16)2

x = 6 ± i√162

x = 6 ± 4i2

x = 3 ± 2i

Answer: x = 3 ± 2i

3
Solve by the quadratic formula. 2x² + 4x + 5 = 0
Show Answer
a = 2
b = 4
c = 5

x = −b ± √(b² − 4ac)2a

x = −4 ± √(4² − 4(2)(5))4

x = −4 ± √(16 − 40)4

x = −4 ± √(−24)4

x = −4 ± i√244

x = −4 ± 2i√64

x = −2 ± i√62

Answer: x = −2 ± i√62

4
Solve by the quadratic formula. 3x² + 2x + 2 = 0
Show Answer
a = 3
b = 2
c = 2

x = −b ± √(b² − 4ac)2a

x = −2 ± √(2² − 4(3)(2))6

x = −2 ± √(4 − 24)6

x = −2 ± √(−20)6

x = −2 ± i√206

x = −2 ± 2i√56

x = −1 ± i√53

Answer: x = −1 ± i√53

5
Solve by the quadratic formula. 2x² − 2x + 5 = 0
Show Answer
a = 2
b = −2
c = 5

x = −b ± √(b² − 4ac)2a

x = −(−2) ± √((−2)² − 4(2)(5))4

x = 2 ± √(4 − 40)4

x = 2 ± √(−36)4

x = 2 ± i√364

x = 2 ± 6i4

x = 1 ± 3i2

Answer: x = 1 ± 3i2

Level 3

Hard

The equation is not set equal to zero. One, two, or zero real solutions.

1
Solve by the quadratic formula. 2x² + 5x = −2
Show Answer

2x² + 5x = −2

2x² + 5x + 2 = 0

a = 2
b = 5
c = 2

x = −b ± √(b² − 4ac)2a

x = −5 ± √(5² − 4(2)(2))4

x = −5 ± √(25 − 16)4

x = −5 ± √94

x = −5 ± 34

x = −5 + 34

x = −24

x = −12

x = −5 − 34

x = −84

x = −2

Answer: x = −12 and x = −2

2
Solve by the quadratic formula. 2x² = 6x − 1
Show Answer

2x² = 6x − 1

2x² − 6x + 1 = 0

a = 2
b = −6
c = 1

x = −b ± √(b² − 4ac)2a

x = −(−6) ± √((−6)² − 4(2)(1))4

x = 6 ± √(36 − 8)4

x = 6 ± √284

x = 6 ± 2√74

x = 3 ± √72

Answer: x = 3 ± √72

3
Solve by the quadratic formula. x² − 4x + 7 = 3
Show Answer

x² − 4x + 7 = 3

x² − 4x + 4 = 0

a = 1
b = −4
c = 4

x = −b ± √(b² − 4ac)2a

x = −(−4) ± √((−4)² − 4(1)(4))2

x = 4 ± √(16 − 16)2

x = 4 ± √02

x = 4 ± 02

x = 42

x = 2

Answer: x = 2

4
Solve by the quadratic formula. x² + 4x + 9 = 2x
Show Answer

x² + 4x + 9 = 2x

x² + 2x + 9 = 0

a = 1
b = 2
c = 9

x = −b ± √(b² − 4ac)2a

x = −2 ± √(2² − 4(1)(9))2

x = −2 ± √(4 − 36)2

x = −2 ± √(−32)2

x = −2 ± i√322

x = −2 ± 4i√22

x = −1 ± 2i√2

Answer: x = −1 ± 2i√2

5
Solve by the quadratic formula. 2x² + 3x = 2
Show Answer

2x² + 3x = 2

2x² + 3x − 2 = 0

a = 2
b = 3
c = −2

x = −b ± √(b² − 4ac)2a

x = −3 ± √(3² − 4(2)(−2))4

x = −3 ± √(9 − (−16))4

x = −3 ± √(9 + 16)4

x = −3 ± √254

x = −3 ± 54

x = −3 + 54

x = 24

x = 12

x = −3 − 54

x = −84

x = −2

Answer: x = 12 and x = −2

Quiz

Quiz

A mix from every level. Use this to see if you are ready.

1
Solve by the quadratic formula. 2x² − 7x + 3 = 0
Show Answer
a = 2
b = −7
c = 3

x = −b ± √(b² − 4ac)2a

x = −(−7) ± √((−7)² − 4(2)(3))4

x = 7 ± √(49 − 24)4

x = 7 ± √254

x = 7 ± 54

x = 7 + 54

x = 124

x = 3

x = 7 − 54

x = 24

x = 12

Answer: x = 3 and x = 12

2
Solve by the quadratic formula. x² + 2x − 4 = 0
Show Answer
a = 1
b = 2
c = −4

x = −b ± √(b² − 4ac)2a

x = −2 ± √(2² − 4(1)(−4))2

x = −2 ± √(4 − (−16))2

x = −2 ± √(4 + 16)2

x = −2 ± √202

x = −2 ± 2√52

x = −1 ± √5

Answer: x = −1 ± √5

3
Solve by the quadratic formula. x² − 10x + 28 = 3
Show Answer

x² − 10x + 28 = 3

x² − 10x + 25 = 0

a = 1
b = −10
c = 25

x = −b ± √(b² − 4ac)2a

x = −(−10) ± √((−10)² − 4(1)(25))2

x = 10 ± √(100 − 100)2

x = 10 ± √02

x = 10 ± 02

x = 102

x = 5

Answer: x = 5

4
Solve by the quadratic formula. x² − 4x + 13 = 0
Show Answer
a = 1
b = −4
c = 13

x = −b ± √(b² − 4ac)2a

x = −(−4) ± √((−4)² − 4(1)(13))2

x = 4 ± √(16 − 52)2

x = 4 ± √(−36)2

x = 4 ± i√362

x = 4 ± 6i2

x = 2 ± 3i

Answer: x = 2 ± 3i

5
Solve by the quadratic formula. 3x² = 4x + 4
Show Answer

3x² = 4x + 4

3x² − 4x − 4 = 0

a = 3
b = −4
c = −4

x = −b ± √(b² − 4ac)2a

x = −(−4) ± √((−4)² − 4(3)(−4))6

x = 4 ± √(16 − (−48))6

x = 4 ± √(16 + 48)6

x = 4 ± √646

x = 4 ± 86

x = 4 + 86

x = 126

x = 2

x = 4 − 86

x = −46

x = −23

Answer: x = 2 and x = −23

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