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Question 1 of 4
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To find the roots of an equation in the form a x 2 + b x + c = 0 a ≠ 0 x = - b ± √ b 2 - 4 a c 2 a
The equation x 2 + 6 x + 13 = 0 a = 1 b = 6 c = 13
x = - b ± √ b 2 - 4 a c 2 a Substitute in a = 1 b = 6 c = 13
x = - 6 ± √ 6 2 - 4 ( 1 ) ( 13 ) 2 ( 1 ) Simplify
x = - 6 ± √ 6 2 - 4 ( 1 ) ( 13 ) 2 Simplify 6 2
x = - 6 ± √ 36 - 4 ( 1 ) ( 13 ) 2 Multiply - 4 ( 1 ) ( 13 )
x = - 6 ± √ 36 - 52 2 Subtract under the root.
x = - 6 ± √ - 16 2 Separate out the √ - 1
x = - 6 ± √ - 1 × √ 16 2 Replace the √ - 1 √ - 1 = i
x = - 6 ± i × √ 16 2 Simplify the root.
x = - 6 ± i × 4 2 Rearrange - i × 4 i
x = - 6 ± 4 i 2 Divide both terms in the numerator by 2
x = - 3 ± 2 i
x = - 3 + 2 i x = - 3 - 2 i Break into the two solutions.
Question 2 of 4
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To find the roots of an equation in the form a x 2 + b x + c = 0 a ≠ 0 x = - b ± √ b 2 - 4 a c 2 a
The equation x 2 + 6 x + 13 = 0 a = 1 b = 2 c = 9
x = - b ± √ b 2 - 4 a c 2 a Substitute in a = 1 b = 2 c = 9
x = - 2 ± √ 2 2 - 4 ( 1 ) ( 9 ) 2 ( 1 ) Simplify
x = - 2 ± √ 2 2 - 4 ( 1 ) ( 9 ) 2 Simplify 2 2
x = - 2 ± √ 4 - 4 ( 1 ) ( 9 ) 2 Multiply - 4 ( 1 ) ( 9 )
x = - 2 ± √ 4 - 36 2 Subtract under the root.
x = - 2 ± √ - 32 2 Separate out the √ - 1
x = - 2 ± √ - 1 √ 32 2 Replace the √ - 1 √ - 1 = i
x = - 2 ± i √ 32 2 Simplify using √ 32 = √ 16 × √ 2
x = - 2 ± i √ 16 × √ 2 2 Simplify the root.
x = - 2 ± i 4 √ 2 2 Rearrange i × 4 × √ 2 i
x = - 2 ± 4 i √ 2 2 Divide both terms in the numerator by 2
x = - 1 ± 2 i √ 2
x = - 1 + 2 i √ 2 x = - 1 - 2 i √ 2 Break into the two solutions.
Question 3 of 4
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To find the roots of an equation in the form a x 2 + b x + c = 0 a ≠ 0 x = - b ± √ b 2 - 4 a c 2 a
The equation x 2 + 6 x + 13 = 0 a = 1 b = 5 c = 25
x = - b ± √ b 2 - 4 a c 2 a Substitute in a = 1 b = 5 c = 25
x = - 5 ± √ 5 2 - 4 ( 1 ) ( 25 ) 2 ( 1 ) Simplify
x = - 5 ± √ 5 2 - 4 ( 1 ) ( 25 ) 2 Simplify 5 2
x = - 5 ± √ 25 - 4 ( 1 ) ( 25 ) 2 Multiply - 4 ( 1 ) ( 25 )
x = - 5 ± √ 25 - 100 2 Subtract under the root.
x = - 5 ± √ - 75 2 Separate out the √ - 1
x = - 5 ± √ - 1 × √ 75 2 Replace the √ - 1 √ - 1 = i
x = - 5 ± i × √ 25 × √ 3 2 Simplify the root using √ 75 = √ 25 × √ 3
x = - 5 ± i × 5 × √ 3 2 Simplify the root.
x = - 5 ± 5 i √ 3 2 Rearrange - i × 4 i
x = - 5 2 ± 5 i √ 3 2 Divide both terms in the numerator by 2
- 5 2 + 5 i √ 3 2 - 5 2 - 5 i √ 3 2 Break into the two solutions.
Question 4 of 4
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To find the roots of an equation in the form a x 2 + b x + c = 0 a ≠ 0 x = - b ± √ b 2 - 4 a c 2 a
Simplify the equation 3 x + 1 x = 2 x 3 x 2 + 1 = 2 x a x 2 + b x + c = 0 3 x 2 - 2 x + 1 = 0
The equation 3 x 2 - 2 x + 1 = 0 a = 3 b = - 2 c = 1
x = - b ± √ b 2 - 4 a c 2 a Substitute in a = 3 b = - 2 c = 1
x = - ( - 2 ) ± √ ( - 2 ) 2 - 4 ( 3 ) ( 1 ) 2 ( 3 ) Simplify - ( - 2 )
x = 2 ± √ ( - 2 ) 2 - 4 ( 3 ) ( 1 ) 2 ( 3 ) Simplify ( - 2 ) 2
x = 2 ± √ 4 - 4 ( 3 ) ( 1 ) 2 ( 3 )
Simplify - 4 ( 3 ) ( 1 )
x = 2 ± √ 4 - 12 2 ( 3 ) Subtract under the root.
x = 2 ± √ - 8 2 ( 3 ) Multiply 2 ( 3 )
x = 2 ± √ - 1 × √ 8 6 Separate out the √ - 1
x = 2 ± i × √ 8 6 Replace the √ - 1 √ - 1 = i
x = 2 ± i × √ 4 × 2 6 Simplify the root.
x = 2 ± i × 2 √ 2 6 Rearrange i × 2 √ 2 i
x = 2 6 ± 2 √ 2 6 i
Divide both terms in the numerator by 2
x = 1 3 ± √ 2 3 i
Simplify
x = 1 3 + √ 2 3 i x = 1 3 - √ 2 3 i Break into the two solutions.
