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: Start by isolating the square root term on one side of the equation. The given equation is 2√(x - 1) = x. Divide both sides of the equation by 2 to isolate √(x - 1), resulting in √(x - 1) = x / 2.
2. Step 2: Eliminate the square root by squaring both sides of the equation. When you square √(x - 1), it becomes x - 1. On the right-hand side, squaring (x / 2) results in (x^2) / 4. The equation now becomes x - 1 = (x^2) / 4.
3. Step 3: Eliminate the fraction by multiplying through by 4 to simplify the equation. Multiply every term by 4, resulting in 4(x - 1) = x^2. Expand the left-hand side to get 4x - 4 = x^2.
4. Step 4: Rearrange the equation into standard quadratic form. Subtract 4x and add 4 to both sides to set the equation equal to 0. This gives x^2 - 4x + 4 = 0.
5. Step 5: Factor the quadratic equation. Notice that x^2 - 4x + 4 is a perfect square trinomial, so it factors as (x - 2)^2 = 0. Solve for x by setting (x - 2) = 0, which gives x = 2. Verify the solution by substituting it back into the original equation to ensure it satisfies the equation.

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 crucial. This includes knowing how to isolate the square root and how to square both sides of an equation to eliminate the square root.

Isolating Variables

Isolating variables is a fundamental algebraic technique used to solve equations. It involves rearranging the equation to get the variable of interest on one side. In this case, isolating x will help in simplifying the equation and finding its value, which is essential for solving the given equation.

Checking Solutions

Checking solutions is the process of substituting the found values back into the original equation to verify their correctness. This step is important because squaring both sides of an equation can introduce extraneous solutions that do not satisfy the original equation. Ensuring that the solution is valid is a critical part of the problem-solving process.