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: \( (3x - 1)^2 = 12 \).
2. Apply the square root property, which states that if \( A^2 = B \), then \( A = \pm \sqrt{B} \). Here, set \( 3x - 1 = \pm \sqrt{12} \).
3. Simplify \( \sqrt{12} \) by expressing it as \( \sqrt{4 \times 3} = 2\sqrt{3} \), so the equation becomes \( 3x - 1 = \pm 2\sqrt{3} \).
4. Solve for \( x \) by isolating it: add 1 to both sides to get \( 3x = 1 \pm 2\sqrt{3} \), then divide both sides by 3 to find \( x = \frac{1 \pm 2\sqrt{3}}{3} \).
5. Write the two solutions explicitly as \( x = \frac{1 + 2\sqrt{3}}{3} \) and \( x = \frac{1 - 2\sqrt{3}}{3} \).

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 solving quadratic equations by isolating the squared term and taking the square root of both sides, introducing both positive and negative roots.

Isolating the Variable

Before applying the square root property, the equation must be manipulated so that the squared expression is isolated on one side. This involves algebraic steps such as dividing or adding/subtracting terms to simplify the equation for easier solving.

Solving Linear Equations After Taking Square Roots

After applying the square root property, the resulting equations are linear and can be solved by isolating the variable. This step involves basic algebraic operations like addition, subtraction, multiplication, or division to find the values of the variable.