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Chapter 7: Problem 10

Solve each equation. $$ c^{2}-20=44 $$

Short Answer

Expert verified
The solutions are \( c = 8 \) and \( c = -8 \).

Step by step solution

01

Isolate the variable term

Add 20 to both sides of the equation to isolate the variable term. \[ c^2 - 20 + 20 = 44 + 20 \] This simplifies to: \[ c^2 = 64 \]
02

Take the square root of both sides

To solve for \( c \), take the square root of both sides of the equation: \[ \sqrt{c^2} = \sqrt{64} \] This results in: \[ c = \pm 8 \]
03

State the solution

The solutions to the equation are \( c = 8 \) and \( c = -8 \).

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Key Concepts

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

Isolating Variables
When solving equations, the first step is often isolating the variable. This means getting the variable term on one side of the equation by itself.
In the given exercise, you have the equation \( c^2 - 20 = 44 \).
To isolate \( c^2 \), you add 20 to both sides:
\[ c^2 - 20 + 20 = 44 + 20 \] Simplifying this, you get:
\[ c^2 = 64 \] Now, the variable term \( c^2 \) is by itself on one side of the equation, which makes it easier to solve.
Square Roots
Once the variable term is isolated, the next step is often to use square roots to solve for the variable.
In our exercise, we have \( c^2 = 64 \).
To find \( c \), we take the square root of both sides:
\[ \sqrt{c^2} = \sqrt{64} \] The square root of \( c^2 \) is \( c \), and the square root of 64 is 8, so:
\[ c = \pm 8 \] It's important to remember that taking the square root of both sides gives us two solutions: a positive and a negative value.
Positive and Negative Solutions
When solving equations involving squares, always remember there are typically two solutions: a positive and a negative.
This is because both a positive and a negative number, when squared, will give the same result.
For example:
  • \( 8^2 = 64 \)
  • \( (-8)^2 = 64 \)
In our exercise, \( c = 8 \) and \( c = -8 \) are both valid solutions.
Always make sure to check for both when solving similar problems!

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Most popular questions from this chapter

Solve each equation by completing the square. $$ x^{2}-6 x=-5 $$

Factor the expression on the left side of each equation as much as possible, and find all the possible solutions. It will help to remember that \(x^{4}=\left(x^{2}\right)^{2}, x^{8}=\left(x^{4}\right)^{2},\) and \(x^{3}=x\left(x^{2}\right) .\) $$ x^{4}-2 x^{2}+1=0 $$

Consider the equation \(3 x+\frac{1}{x}=4\) a. Do you see any obvious solutions to this equation? b. Now solve the equation using the quadratic formula. (Hint: First write an equivalent quadratic equation.) Check your solutions in the original equation.

Factor the expression on the left side of each equation as much as possible, and find all the possible solutions. It will help to remember that \(x^{4}=\left(x^{2}\right)^{2}, x^{8}=\left(x^{4}\right)^{2},\) and \(x^{3}=x\left(x^{2}\right) .\) $$ x^{8}-1=0 $$

Expand each expression. $$ (2 e-4)(e-6) $$
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