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Chapter 5: Problem 5

Solve. $$(x-4)(x+1)=0$$

Short Answer

Expert verified
The solutions to the equation \((x-4)(x+1) = 0\) are \(x = 4\) and \(x = -1\).

Step by step solution

01

Apply Zero-Product Property

Given quadratic equation is \((x-4)(x+1) = 0\). According to the zero-product property, if the product of multiple factors is zero, then at least one of the factors must be zero. Therefore, set each factor equal to zero: \((x-4) = 0\) and \((x+1) = 0\).
02

Solve for x in Each Equation

Solving the two separate equations for x. For the first equation, add 4 to both sides: \(x - 4 + 4 = 0 + 4\), which simplifies to \(x = 4\). Now, for the second equation, subtract 1 from both sides: \(x + 1 - 1 = 0 - 1\), which simplifies to \(x = -1\).
03

Confirm the Solutions

The potential solutions for the equation, according to the zero-product property, are \(x = 4\) and \(x = -1\). Plug these values back into the original equation to confirm. For \(x = 4\), \((4-4)(4+1) = 0\). And for \(x = -1\), \((-1-4)(-1+1) = 0\). Both these tests yield 0, confirming that \(x = 4\) and \(x = -1\) are the correct solutions.

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