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Chapter 5: Problem 1

Solve. $$(x-8)(x-7)=0$$

Short Answer

Expert verified
The solutions to the equation are x = 8 and x = 7.

Step by step solution

01

Set each factor to zero

For a product of factors to be zero, at least one of the factors has to be zero. Thus, set each factor of the given equation equal to zero, which means (x - 8) = 0 and (x - 7) = 0.
02

Solve each equation

When we solve each equation from step 1, we obtain the values of x. Therefore, solving (x - 8) = 0, we will have x = 8. Also, solving (x - 7) = 0, we have x = 7.

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Key Concepts

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

Understanding Factoring in Quadratic Equations
Factoring is a key concept in solving quadratic equations. It involves writing the equation as a product of its factors. For example, in the equation \((x-8)(x-7)=0\), the expression \((x-8)\) and \((x-7)\) are the factors.

You'll frequently encounter quadratic equations in the form \(ax^2 + bx + c = 0\). The goal is to express this as \((px+q)(rx+s)=0\). When an equation is written as a product of factors, it becomes easier to solve using other methods.

Key points to consider when factoring:
  • Identify common factors first.
  • Look for patterns, such as the difference of squares.
  • Apply methods like grouping if necessary.
Factoring transforms equations into simpler forms, making them manageable and solvable.
The Zero Product Property Explained
The Zero Product Property is an important principle in algebra. It states that if a product of two expressions equals zero, then at least one of the expressions must be zero.

In the equation \((x-8)(x-7)=0\), we apply the Zero Product Property to find the solutions. This means setting each factor equal to zero and solving for \(x\).

Why is this property so useful?
  • It simplifies solving equations by breaking them into individual parts.
  • It directly leads to potential solutions without complex calculations.
Remember, the Zero Product Property only applies when the product equals zero. This step is a vital transition from factoring to extracting solutions.
Solving Quadratic Equations with Ease
Solving quadratic equations can sound challenging, but with the right techniques, it becomes straightforward. After you've factored the equation and applied the Zero Product Property, solving becomes direct.

For example, given \((x-8)=0\) and \((x-7)=0\) from the original problem, simply find \(x\) values that make these equations true:
  • For \((x-8)=0\), add 8 to both sides to get \(x=8\).
  • For \((x-7)=0\), add 7 to both sides to get \(x=7\).
This method ensures you find the right solutions with minimal effort.

Here are some general tips for solving quadratic equations:
  • Always check if the equation can be factored.
  • Don't forget special cases like perfect squares or simple zeroes.
With practice, you'll find solving these equations becomes second nature.

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Most popular questions from this chapter

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Factor. $$x^{2}+4 x+4-y^{2}$$
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