Problem 82 - Introductory Algebra for College Students

Question

Explain how to solve \((x-1)^{2}=16\) using the square root property.

Step-by-Step Solution

Answer

The solutions to the equation \((x-1)^{2}=16\) are \(x=5\) and \(x=-3\).

Step 1: Isolate the squared term

In the given equation the squared term is already isolated, i.e., \((x-1)^{2}=16\).

Step 2: Square Root Property

Now,apply the square root property by taking the square root of both sides. This gives us \(x-1 = \sqrt{16}\) and \(x-1 = -\sqrt{16}\). Remember, when we take the square root of both sides we need to consider both positive and negative roots.

Step 3: Simplify Square Roots

Now simplify the square root of 16. The square root of 16 is 4. So, the equations become \(x-1=4\) and \(x-1=-4\).

Step 4: Solve for x

To isolate x, add 1 to both sides in each equation. This gives two solutions: \(x=4+1=5\) and \(x=-4+1=-3\).