Solve $(v-5)^2=6$ where $v$ is a real number. Simplify your answer as much as possible. (If there is more than one solution, separate them with commas.)
$v=$
Answer (1)
Take the square root of both sides: v − 5 = ± 6 .
Isolate v : v = 5 ± 6 .
The solutions are 5 + 6 and 5 − 6 .
The final answer is 5 + 6 , 5 − 6 .
Explanation
Problem Analysis We are given the equation ( v − 5 ) 2 = 6 and we need to solve for v .
Taking the Square Root To solve for v , we first take the square root of both sides of the equation: ( v − 5 ) 2 = 6 ( v − 5 ) 2 = ± 6 v − 5 = ± 6
Isolating v Next, we isolate v by adding 5 to both sides of the equation: v = 5 ± 6
Finding the Solutions Therefore, the two solutions for v are 5 + 6 and 5 − 6 .
Final Answer The solutions are v = 5 + 6 , 5 − 6 .
Examples
Understanding how to solve quadratic equations like this is useful in many real-world situations. For example, if you are designing a square garden and need to determine the length of each side to achieve a specific area, you might end up solving a similar equation. Also, in physics, when calculating the time it takes for an object to fall from a certain height, you often encounter quadratic equations. Knowing how to solve them allows you to find the unknown variable, such as the side length of the garden or the time of the fall.
