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Chapter 7: Problem 18

In Exercises \(1-21,\) solve the equation for the variable. $$ \sqrt{r^{2}+144}=13 $$

Short Answer

Expert verified
Answer: The two possible values for r are 5 and -5.

Step by step solution

01

Write down the given equation

The given equation is: $$ \sqrt{r^{2}+144}=13 $$
02

Square both sides of the equation

To remove the square root, we need to square both sides. This is because squaring a square root will cancel the square root. We get: $$ (\sqrt{r^{2}+144})^{2} = 13^{2} $$
03

Simplify the equation

Now we can simplify the equation by canceling the square root and computing the square of 13: $$ r^{2} + 144 = 169 $$
04

Isolate the variable r

Now we need to isolate the variable r by subtracting 144 from both sides: $$ r^{2} = 169 - 144 $$ $$ r^{2} = 25 $$
05

Solve for r

Now we need to find the square root of both sides of the equation to solve for r. Note that there will be two solutions: one positive and one negative. So, we have: $$ \sqrt{r^{2}} = \sqrt{25} $$ \( r = 5\) or \( r = -5\) The solutions to the given equation are \(r = 5\) and \(r = -5\).

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Key Concepts

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

Understanding Square Roots
A square root is a number that, when multiplied by itself, yields the original number. In equations, the square root symbol \( \sqrt{} \) represents this concept. For example, \( \sqrt{16} = 4 \) because \( 4 \times 4 = 16 \). In the context of solving equations like \( \sqrt{r^{2}+144}=13 \), the square root presents a key step in unraveling the variable. It's important to recognize that every square root operation can potentially yield two results: a positive and a negative value.
  • Square roots are reversed by squaring the number.
  • Every positive root will have a mirrored negative root.
  • Understanding square roots helps in predicting the possible solutions.
Isolate the Variable
When solving equations, one primary goal is to "isolate the variable". This means rearranging the equation until the unknown variable鈥攖he letter representing the value you want to find鈥攊s alone on one side of the equation. Isolating the variable simplifies complex equations and narrows down the possible solutions. In our example, the equation \( r^{2} + 144 = 169 \) requires us to get \( r \) by itself. First, subtract 144 from both sides to begin stripping away unnecessary elements around \( r \).
  • Subtract or add terms to both sides as needed.
  • Focus on operations that will cancel out added terms.
  • Work step-by-step towards simplifying the expression.
How to Square Both Sides
Squaring both sides of an equation is a powerful move to eliminate a square root and make it easier to isolate the variable. In our initial problem, we started with \( \sqrt{r^{2}+144}=13 \). By squaring both sides, we effectively canceled out the square root, transforming it into a simpler form \((r^{2} + 144 = 169)\). This algebraic technique works because it applies equal effort to both sides, maintaining balance. However, remember it affects both the magnitude and direction of potential solutions.
  • Balance is key; apply squaring uniformly to both sides.
  • Squaring transforms the equation to an easier-to-handle polynomial.
  • Take care with squaring to preserve equation integrity.

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Most popular questions from this chapter

The equation $$ x \sqrt{8-x}=5 $$ has two solutions. Are they positive, zero, or negative? Give an algebraic reason why this must be the case. You need not find the solutions.

In Problems \(59-70,\) decide for what values of the constant \(A\) the equation has (a) The solution \(t=0\) (b) A positive solution (c) A negative solution $$ (-t)^{4}=A $$

If the radius of a circle is halved, what happens to its area?

In Problems \(59-70,\) decide for what values of the constant \(A\) the equation has (a) The solution \(t=0\) (b) A positive solution (c) A negative solution $$ t^{4}=A^{2} $$

If \(z\) is proportional to a power of \(y\) and \(y\) is proportional to a power of \(x\), is \(z\) proportional to a power of \(x\) ?
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