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Step-by-step explanation:
1. Expand the left side using distribution:
2x(x + 1)
Distribute 2x across x + 1:
2x * x + 2x * 1
= 2x^2 + 2x
2. Rewrite the equation with the expanded form:
2x^2 + 2x = -8
3. Move all terms to one side to set the equation to zero:
2x^2 + 2x + 8 = 0
4. Solve the quadratic equation 2x^2 + 2x + 8 = 0 using the quadratic formula:
The quadratic formula is x = (-b ± sqrt(b^2 - 4ac)) / (2a)
Here, a = 2, b = 2, and c = 8
- Calculate the discriminant:
Discriminant = b^2 - 4ac
= 2^2 - 4 * 2 * 8
= 4 - 64
= -60
- Compute the roots using the quadratic formula:
x = (-b ± sqrt(Discriminant)) / (2a)
x = (-2 ± sqrt(-60)) / (2 * 2)
x = (-2 ± sqrt(-1 * 60)) / 4
sqrt(-60) = sqrt(-1 * 60) = i * sqrt(60) = i * sqrt(4 * 15) = 2i * sqrt(15)
x = (-2 ± 2i * sqrt(15)) / 4
x = (-1 ± i * sqrt(15)) / 2
Final Answer:
The solutions are:
x = (-1 + i * sqrt(15)) / 2
and
x = (-1 - i * sqrt(15)) / 2