Answer:
x = -4 + 3 i or x = -4 - 3 i
Explanation:
Solve for x:
(x + 4)/(x + 5) = (x - 5)/(2 x)
Hint: | Multiply both sides by a polynomial to clear fractions.
Cross multiply:
2 x (x + 4) = (x - 5) (x + 5)
Hint: | Write the quadratic polynomial on the left hand side in standard form.
Expand out terms of the left hand side:
2 x^2 + 8 x = (x - 5) (x + 5)
Hint: | Write the quadratic polynomial on the right hand side in standard form.
Expand out terms of the right hand side:
2 x^2 + 8 x = x^2 - 25
Hint: | Move everything to the left hand side.
Subtract x^2 - 25 from both sides:
x^2 + 8 x + 25 = 0
Hint: | Using the quadratic formula, solve for x.
x = (-8 ± sqrt(8^2 - 4×25))/2 = (-8 ± sqrt(64 - 100))/2 = (-8 ± sqrt(-36))/2:
x = (-8 + sqrt(-36))/2 or x = (-8 - sqrt(-36))/2
Hint: | Express sqrt(-36) in terms of i.
sqrt(-36) = sqrt(-1) sqrt(36) = i sqrt(36):
x = (-8 + i sqrt(36))/2 or x = (-8 - i sqrt(36))/2
Hint: | Simplify radicals.
sqrt(36) = sqrt(4×9) = sqrt(2^2×3^2) = 2×3 = 6:
x = (-8 + i×6)/2 or x = (-8 - i×6)/2
Hint: | Factor the greatest common divisor (gcd) of -8, 6 i and 2 from -8 + 6 i.
Factor 2 from -8 + 6 i giving -8 + 6 i:
x = 1/2-8 + 6 i or x = (-8 - 6 i)/2
Hint: | Cancel common terms in the numerator and denominator.
(-8 + 6 i)/2 = -4 + 3 i:
x = -4 + 3 i or x = (-8 - 6 i)/2
Hint: | Factor the greatest common divisor (gcd) of -8, -6 i and 2 from -8 - 6 i.
Factor 2 from -8 - 6 i giving -8 - 6 i:
x = -4 + 3 i or x = 1/2-8 - 6 i
Hint: | Cancel common terms in the numerator and denominator.
(-8 - 6 i)/2 = -4 - 3 i:
Answer: x = -4 + 3 i or x = -4 - 3 i