Solve each quadratic equation using the square root property. See Example 6. (3x - 1)² = 12

Verified step by step guidance

1. Start with the given equation: \( (3x - 1)^2 = 12 \). The goal is to solve for \(x\) by using the square root property.
2. Apply the square root property, which states that if \(A^2 = B\), then \(A = \pm \sqrt{B}\). Here, \(A = 3x - 1\) and \(B = 12\), so write: \( 3x - 1 = \pm \sqrt{12} \).
3. Simplify the square root if possible. Since \(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\), rewrite the equation as: \( 3x - 1 = \pm 2\sqrt{3} \).
4. Isolate the term with \(x\) by adding 1 to both sides: \( 3x = 1 \pm 2\sqrt{3} \).
5. Finally, solve for \(x\) by dividing both sides by 3: \( x = \frac{1 \pm 2\sqrt{3}}{3} \). This gives the two possible solutions for \(x\).

Key Concepts

Square Root Property

The square root property states that if an equation is in the form (ax + b)² = c, then ax + b = ±√c. This allows you to solve quadratic equations by isolating the squared term and taking the square root of both sides, considering both positive and negative roots.

Isolating the Variable

Before applying the square root property, you must isolate the squared expression on one side of the equation. This involves algebraic manipulation such as adding, subtracting, multiplying, or dividing terms to simplify the equation to the form (expression)² = constant.

Solving Linear Equations After Taking Square Roots

After applying the square root property, you get two linear equations due to the ± sign. Solving these linear equations involves isolating the variable to find the two possible solutions for the original quadratic equation.