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Chapter 0: Problem 151

Explain how to solve \(x^{2}+6 x+8=0\) using factoring and the zero-product principle.

Short Answer

Expert verified
The solutions to the equation \(x^{2}+6x+8=0\) are \(x=-2\) and \(x=-4\).

Step by step solution

01

Factor the quadratic equation

First, we need to factor the equation \(x^{2}+6x+8=0\). When factoring quadratics, we're looking for two numbers that add up to the coefficient of the \(x\) term (which is 6) and multiply to the constant term (which is 8). These numbers are 2 and 4, because 2 + 4 = 6 and 2 * 4 = 8. Therefore, the factored form of the equation is \((x+2)(x+4)=0\).
02

Apply the Zero-Product Principle

Next, apply the zero-product principle, which states that if the product of two factors is zero, then at least one of the factors must be zero. Thus we set each factor equal to zero and solve for \(x\): \(x+2=0\) and \(x+4=0\). Solving each of these gives \(x=-2\) and \(x=-4\), respectively.

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