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Chapter 9: Problem 6

Solve each equation. $$v^{2}-80=0$$

Short Answer

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The solutions to

Step by step solution

01

Isolate the quadratic term

To isolate the term with the variable, add 80 to both sides of the equation:
02

Take the square root of both sides

To solve for v, take the square root of both sides of the equation:
03

Simplify the result

Simplify the result to find the values of v:

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Key Concepts

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

Isolating the Variable
When solving quadratic equations, the first step is to isolate the variable term. In the given problem, we start with the equation \( v^2 - 80 = 0 \). To isolate \( v^2 \), we add 80 to both sides of the equation. This step is crucial because it helps us to work with a simpler equation.
Here's what it looks like:
\[ v^2 - 80 + 80 = 0 + 80 \]
This simplifies to:
\[ v^2 = 80 \]
By isolating the quadratic term, we make it easier to solve for the variable in the subsequent steps.
Taking Square Roots
Once we have isolated the variable term, we need to solve for the variable itself. The next key step is to take the square root of both sides of the equation. In the simplified equation \( v^2 = 80 \), we take the square root of both sides to find the value of \( v \).
This looks like:
\[ v = \pm \sqrt{80} \]
In this case:
  • The square root operation introduces both a positive and a negative solution because both \( (\sqrt{80})^2 \) and \( (-\sqrt{80})^2 \) equal 80.
Taking the square root helps us find the potential values of the variable we're solving for.
Simplifying Results
After taking the square root, the final step is to simplify the result. Simplification often helps in expressing the answer in its most reduced form. In our problem, \( v = \pm \sqrt{80} \), we can simplify the square root of 80 further.
Here's how:
  • Use the fact that \( \sqrt{80} = \sqrt{16 \times 5} \).
  • Simplify this to \( \sqrt{16} \times \sqrt{5} \).
  • Since \( \sqrt{16} = 4 \), we get \( v = \pm 4\sqrt{5} \).
Therefore, the simplified solutions to the equation \( v^2 - 80 = 0 \) are:
\[ v = 4\sqrt{5} \] and \[ v = -4\sqrt{5} \].
Simplifying results ensures that we express our answers in the most understandable and conventional form.

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