Problem 12 Solve each equation, and check t... [FREE SOLUTION]

Chapter 5: Problem 12

Solve each equation, and check the solutions. $$ (x-1)(x+8)=0 $$

Short Answer

Expert verified

The solutions are x = 1 and x = -8.

Step by step solution

Apply the Zero Product Property

Use the Zero Product Property which states that if the product of two factors is zero, then at least one of the factors must be zero. Set each factor equal to zero: (1) (x-1) = 0 (2) (x+8) = 0

Solve Each Individual Equation

Solve the equations obtained in Step 1. For (x-1) = 0 Add 1 to both sides: (x = 1) For (x+8) = 0 Subtract 8 from both sides: (x = -8)

Verify the Solutions

Check that each solution satisfies the original equation. Substitute the values back into the original equation For x = 1: (1 - 1)(1 + 8) = 0 (0)(9) = 0, which is True For x = -8: (-8 - 1)(-8 + 8) = 0 (-9)(0) = 0, which is True Both solutions satisfy the original equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Zero Product Property

The Zero Product Property is a fundamental concept in algebra that helps us solve equations involving products of terms set equal to zero. Simply put, if you have a product of two or more terms that equals zero, then at least one of those terms must be zero. For example, if you have the equation $(x-1)(x+8) = 0$, the Zero Product Property tells us that either $x-1 = 0$ or $x+8 = 0$ (or both). This allows us to break the problem into simpler, more manageable parts.

Factoring

Factoring is a technique used to rewrite a polynomial as a product of its factors. When we solve quadratic equations, factoring can often simplify the equation significantly. In the exercise, the quadratic equation $(x-1)(x+8) = 0$ has already been factored for us. The factors here are $(x-1)$ and $(x+8)$. Factoring transforms the equation into a form that allows us to use the Zero Product Property effectively, by focusing on each factor individually instead of the entire equation.

Solution Verification

After finding the potential solutions to an equation, it's crucial to verify that these solutions are correct. This step ensures our work is accurate and that we have not made any mistakes along the way. In our example problem, we found the solutions $x = 1$ and $x = -8$.
To verify these solutions, substitute them back into the original equation $(x-1)(x+8) = 0$. For \x = 1\: $(1-1)(1+8) = (0)(9) = 0$, which is true. For \x = -8\: $(-8-1)(-8+8) = (-9)(0) = 0$, which is also true. Therefore, both solutions are correct and satisfy the original equation.

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91影视

Solve each equation, and check the solutions. $$ (x-1)(x+8)=0 $$

Short Answer

Step by step solution

Apply the Zero Product Property

Solve Each Individual Equation

Verify the Solutions

Key Concepts

Zero Product Property

Factoring

Solution Verification

One App. One Place for Learning.

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