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Chapter 9: Problem 40

\(w^{2}+35=99\)

Short Answer

Expert verified
w = \pm 8

Step by step solution

01

- Isolate the variable term

First, subtract 35 from both sides of the equation to isolate the term with the variable. This gives:\[ w^{2} + 35 - 35 = 99 - 35 \]Simplifying this equation yields:\[ w^{2} = 64 \]
02

- Take the square root of both sides

To solve for \( w \), take the square root of both sides of the equation. Remember that taking the square root of a number can yield both a positive and a negative result:\[ w = \pm \sqrt{64} \]Calculating the square root of 64, we find:\[ w = \pm 8 \]

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Key Concepts

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

Isolate the Variable
To solve equations, we often need to isolate the variable. This means getting the variable, like \( w \), by itself on one side of the equation. Let's look at the problem:

Given: \( w^{2} + 35 = 99 \)

Step 1 is to subtract 35 from both sides to isolate \( w^2 \). Why do we do this? We want to move all constants to the other side. By subtracting 35 from both sides, the equation becomes:

\[ w^{2} = 99 - 35 \]
Simplifying the right side, we get:
\[ w^{2} = 64 \]
Now, the variable term is isolated. The variable \( w \) is now the main focus for further solving. Breaking down large equations into simpler parts by moving constants is key here.
Square Root
Once the variable term is isolated, you'll often need to take the square root to solve for the variable. Here, we have isolated \( w^{2} \):

\[ w^{2} = 64 \]
Taking the square root of both sides helps us find the value of \( w \). Recall that the square of any number has both a positive and negative root. Therefore:

\[ w = \pm \sqrt{64} \]
Here, \( \sqrt{64}=8 \). Notice this means:

\[ w = 8 \] and \[ w = -8 \]
Hence, there are two possible solutions. Always remember the positive and negative possibilities when taking square roots. It can seem confusing, but it鈥檚 very important!
Positive and Negative Solutions
Quadratic equations, like our problem, often yield two solutions. Why? Because both positive and negative numbers squared give the same result. When you take the square root of a positive number, you always get a positive and negative output. Using our example:

Given: \( w^{2} = 64 \), the square root gives:
\[ w = +8 \] and \[ w = -8 \]
So our full set of solutions is:
  • \( w = 8 \)
  • \( w = -8 \)

The final solution includes both. Recognizing both solutions in your work ensures you cover all possible answers. Make sure always to check for both positive and negative roots when working on quadratic equations. This will make your solving process complete and accurate.

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