Question

given \\((x - 7)^2 = 36\\), select the values of \\(x\\).

\\(x = 13\\)
\\(x = 1\\)
\\(x = -29\\)
\\(x = 42\\)

Explanation:

🆕 New Concept Discovered: Solving Quadratic Equations by Taking Square Roots
Undoing a square with positive and negative roots

Step 1: Take the square root of both sides

To isolate the term inside the parentheses, take the square root of both sides of the equation. Remember that a positive number has both a positive and a negative square root:

$$ \sqrt{(x - 7)^2} = \pm\sqrt{36} $$

$$ x - 7 = \pm 6 $$

This gives us two separate linear equations to solve.

Step 2: Solve the first equation (positive root)

Set the expression equal to positive \(6\):

$$ x - 7 = 6 $$

Add \(7\) to both sides:

$$ x = 6 + 7 $$

$$ x = 13 $$

Step 3: Solve the second equation (negative root)

Set the expression equal to negative \(-6\):

$$ x - 7 = -6 $$

Add \(7\) to both sides:

$$ x = -6 + 7 $$

$$ x = 1 $$

Answer:

• \(x = 13\)
• \(x = 1\)