To solve the equation [tex]\( v^2 = a^2 + 2x \)[/tex] for [tex]\( v \)[/tex], we will follow these steps:
1. Understand the Problem:
We are given the equation [tex]\( v^2 = a^2 + 2x \)[/tex] and need to solve for the variable [tex]\( v \)[/tex].
2. Isolate the Variable [tex]\( v \)[/tex]:
The equation is already nicely arranged with [tex]\( v^2 \)[/tex] isolated on one side of the equation. Here is the equation again:
[tex]\[
v^2 = a^2 + 2x
\][/tex]
3. Take the Square Root of Both Sides:
To solve for [tex]\( v \)[/tex], we take the square root of both sides of the equation. Remember that taking the square root introduces both the positive and negative roots. Therefore, we have:
[tex]\[
v = \pm \sqrt{a^2 + 2x}
\][/tex]
4. Write the Solutions:
This gives us two possible solutions for [tex]\( v \)[/tex]:
[tex]\[
v = \sqrt{a^2 + 2x} \quad \text{and} \quad v = -\sqrt{a^2 + 2x}
\][/tex]
5. Summary of the Results:
The solutions to the equation [tex]\( v^2 = a^2 + 2x \)[/tex] are:
[tex]\[
v = -\sqrt{a^2 + 2x} \quad \text{and} \quad v = \sqrt{a^2 + 2x}
\][/tex]
We identified both the positive and negative solutions for [tex]\( v \)[/tex] when solving the equation [tex]\( v^2 = a^2 + 2x \)[/tex]. Therefore, the final answer includes both [tex]\( \sqrt{a^2 + 2x} \)[/tex] and [tex]\(-\sqrt{a^2 + 2x} \)[/tex].
