Given $(x-7)^2=36$, select the values of $x$. Choose two correct answers.
$x=-29$
$x=42$
$x=1$
$x=13$
Answer (1)
Take the square root of both sides of the equation: x − 7 = ± 36 .
Simplify the square root: x − 7 = ± 6 .
Solve for x in both cases: x = 7 + 6 = 13 and x = 7 − 6 = 1 .
The solutions are x = 1 , x = 13 .
Explanation
Understanding the Problem We are given the equation ( x − 7 ) 2 = 36 and asked to find the values of x that satisfy this equation. We will use the square root property to solve for x .
Applying the Square Root Property Taking the square root of both sides of the equation ( x − 7 ) 2 = 36 , we get x − 7 = ± 36 .
Simplifying the Square Root Since 36 = 6 , we have x − 7 = ± 6 . This gives us two equations to solve: x − 7 = 6 and x − 7 = − 6 .
Solving for x (Case 1) For the first equation, x − 7 = 6 , we add 7 to both sides to get x = 6 + 7 = 13 .
Solving for x (Case 2) For the second equation, x − 7 = − 6 , we add 7 to both sides to get x = − 6 + 7 = 1 .
Final Solutions Therefore, the two solutions are x = 13 and x = 1 .
Examples
Understanding how to solve quadratic equations by using the square root property is a fundamental skill in algebra. For example, if you are designing a square garden and know the area, you can use this method to find the length of each side. If the area of the garden is represented by ( x − 2 ) 2 = 25 , where x is related to the side length, solving for x will tell you the dimensions you need to build your garden.
