/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 2 Solve each equation. $$ (r-8... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 2

Solve each equation. $$ (r-8)^{2}+3=52 $$

Short Answer

Expert verified
r = 15 or r = 1.

Step by step solution

01

Isolate the squared term

Start by isolating the squared term \( (r-8)^{2} \) on one side of the equation. Subtract 3 from both sides to get: \ (r-8)^{2} + 3 - 3 = 52 - 3 \ which simplifies to \ (r-8)^{2} = 49 \.
02

Take the square root of both sides

Take the square root of both sides of the equation to solve for \( r-8 \). This gives us: \ \sqrt{(r-8)^{2}} = \sqrt{49} \ which simplifies to \ r-8 = \pm 7 \.
03

Solve for r

Solve for \( r \) by adding 8 to both sides of each equation: \ r - 8 = 7 \ and \ r - 8 = -7 \. This results in: \ r = 15 \ or \ r = 1 \.

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Key Concepts

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

Isolating the Variable
In solving quadratic equations, one essential step is isolating the variable. This means getting the term with the variable all by itself on one side of the equation. Let's consider the equation \((r-8)^{2}+3=52\). Our goal here is to isolate the term \((r-8)^{2}\).

To do this, we'll need to eliminate the constant term on the same side as the variable term. In this case, we subtract 3 from both sides of the equation:

\[(r-8)^{2} + 3 - 3 = 52 - 3\]

This simplifies to:

\[(r-8)^{2} = 49\]

By isolating the squared term, we make it easier to work with the equation and solve for the variable. This process sets up the next step, which is taking the square root of both sides.
Taking the Square Root
Once the variable term is isolated, the next step involves taking the square root of both sides of the equation. This step is crucial for removing the exponent and simplifying the expression. Let's apply this to our example from the previous section:

\[(r-8)^{2} = 49\]

Taking the square root of both sides, we get:

\[\sqrt{(r-8)^{2}} = \sqrt{49}\]

It simplifies to:

\[r-8 = \pm 7\]

Notice the \(\pm 7\). The square root of 49 is both 7 and -7 because both \(7^{2}\) and \((-7)^{2}\) result in 49. This gives us two potential solutions for \(r-8\). It's essential to remember this step to ensure we find all possible solutions.
Solving for the Variable
The final step is solving for the variable itself. We have two potential equations from the square root step:

\[r - 8 = 7\]

and

\[r - 8 = -7\]

Let's solve each one by adding 8 to both sides:
  • From \ r - 8 = 7 \:
    \[r = 7 + 8\]
    \[r = 15\]
  • From \ r - 8 = -7 \:
    \[r = -7 + 8\]
    \[r = 1\]
So, the solutions to the equation \ (r-8)^{2}+3=52 \ are \ r = 15 \ and \ r = 1 \.

Note that solving for the variable often involves straightforward arithmetic. Adding or subtracting from both sides will isolate the variable. This step solidifies the final solutions, ensuring that both potential values are considered.

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