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Chapter 5: Problem 8

Solve each equation. $$ (x+7)(x+3)=0 $$

Short Answer

Expert verified
The solutions are x = -7 and x = -3.

Step by step solution

01

Apply Zero Product Property

According to the Zero Product Property, if a product of two factors equals zero, then at least one of the factors must be zero. Therefore, set each factor equal to zero: \(x + 7 = 0\) and \(x + 3 = 0\).
02

Solve the First Equation

Solve the equation \(x + 7 = 0\). Subtract 7 from both sides to isolate the variable x: \(x = -7\).
03

Solve the Second Equation

Solve the equation \(x + 3 = 0\). Subtract 3 from both sides to isolate the variable x: \(x = -3\).
04

Combine the Solutions

The solutions to the original equation are \(x = -7\) and \(x = -3\).

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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 key concept in algebra, especially when solving quadratic equations. It states that if the product of two or more factors equals zero, then at least one of the factors must be zero. This is crucial when dealing with equations like \((x+7)(x+3)=0\). To solve this, we set each factor to zero separately:
  • For the first factor: \(x + 7 = 0\)
  • For the second factor: \(x + 3 = 0\)
By solving these simpler equations, we find the values of \(x\) that satisfy the original equation.
It's important to remember that the Zero Product Property only works when the equation is set to zero. Always ensure your equation is in the form of \(A \cdot B = 0\) before applying this property.
Factoring
Factoring is another essential concept when working with quadratic equations. Factoring involves breaking down a complex expression into simpler parts or 'factors' that, when multiplied together, produce the original expression.
In our exercise, we start with the factored equation \((x+7)(x+3)=0\). Sometimes, you might need to factor a more complex quadratic expression like \(ax^2 + bx + c\).
Here鈥檚 a quick refresher:
  • Look for common factors first; if there are none, proceed to other factoring techniques like grouping or using the quadratic formula.
  • Rewrite the equation in product form, ensuring it equates to zero.
  • Apply the Zero Product Property to set each factor to zero and solve.
Factoring simplifies the problem, making it easier to find the solutions.
Isolating the Variable
Isolating the variable is about rearranging an equation to get the variable alone on one side. This process is fundamental when solving any type of equation.
In our example, after applying the Zero Product Property, we had two simpler equations: \(x + 7 = 0\) and \(x + 3 = 0\).
Here's the step-by-step process to isolate \(x\):
  • For \(x + 7 = 0\), subtract 7 from both sides: \(x = -7\)
  • For \(x + 3 = 0\), subtract 3 from both sides: \(x = -3\)
By isolating \(x\), we make it clear what values \(x\) must take to satisfy each equation. This method is a powerful tool in algebra, allowing you to systematically determine the solutions.

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