(x%5E2-4x%2B16)%3D0.jpeg "(x+4)(x^2-4x+16)=0")
Solving the Equation (x+4)(x^2-4x+16) = 0
This equation is already factored, making it relatively straightforward to solve. Here's how we can approach it:
Understanding the Equation
The equation represents the product of two factors:
- (x+4): This is a linear factor.
- (x^2-4x+16): This is a quadratic factor.
For the product of these factors to equal zero, at least one of the factors must be equal to zero.
Solving for x
Let's apply the zero product property:
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Set each factor equal to zero:
- x + 4 = 0
- x^2 - 4x + 16 = 0
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Solve the linear equation:
- x = -4
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Solve the quadratic equation:
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This quadratic equation doesn't factor easily, so we can use the quadratic formula:
- x = [-b ± √(b^2 - 4ac)] / 2a
- Where a = 1, b = -4, and c = 16
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Substituting the values:
- x = [4 ± √((-4)^2 - 4 * 1 * 16)] / 2 * 1
- x = [4 ± √(-48)] / 2
- x = [4 ± 4√(-3)] / 2
- x = 2 ± 2√(-3)
- x = 2 ± 2i√3 (where 'i' is the imaginary unit, √-1)
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Solutions
Therefore, the solutions to the equation (x+4)(x^2-4x+16) = 0 are:
- x = -4
- x = 2 + 2i√3
- x = 2 - 2i√3
The equation has one real solution (x = -4) and two complex solutions (x = 2 + 2i√3 and x = 2 - 2i√3).