Solve each equation in Exercises 96–102 by the method of your choice. 2√(x-1) = x

Verified step by step guidance

1. Step 1: First, let's isolate the square root on one side of the equation. The equation is already in this form: 2√(x-1) = x.
2. Step 2: To get rid of the square root, we can square both sides of the equation. This gives us (2√(x-1))^2 = x^2.
3. Step 3: Simplify the left side of the equation. Squaring the 2 gives us 4, and squaring the square root of (x-1) gives us (x-1). So, we have 4(x-1) = x^2.
4. Step 4: Expand the left side of the equation to get 4x - 4 = x^2.
5. Step 5: Rearrange the equation to set it equal to zero. This gives us x^2 - 4x + 4 = 0. This is a quadratic equation, which can be solved using the quadratic formula, factoring, or completing the square.

Key Concepts

Square Roots

Square roots are the values that, when multiplied by themselves, yield the original number. In the equation 2√(x-1) = x, understanding how to manipulate square roots is essential. This includes knowing how to isolate the square root and square both sides of the equation to eliminate the root.

Isolating Variables

Isolating variables involves rearranging an equation to get the variable of interest on one side. In this case, you would want to isolate x to solve the equation. This process often requires using inverse operations, such as addition, subtraction, multiplication, or division, to simplify the equation step by step.

Checking Solutions

After solving an equation, it is crucial to check the solutions by substituting them back into the original equation. This step ensures that the solutions are valid and do not introduce extraneous solutions, especially when dealing with square roots, which can sometimes lead to false results if not verified.