Solve each equation in Exercises 15–34 by the square root property. 3(x - 4)^2 = 15

Verified step by step guidance

1. <Step 1: Isolate the squared term.> Divide both sides of the equation by 3 to isolate the squared term: \( (x - 4)^2 = 5 \).
2. <Step 2: Apply the square root property.> Take the square root of both sides of the equation: \( x - 4 = \pm \sqrt{5} \).
3. <Step 3: Solve for x.> Add 4 to both sides of the equation to solve for \( x \): \( x = 4 \pm \sqrt{5} \).
4. <Step 4: Express the solutions.> The solutions are \( x = 4 + \sqrt{5} \) and \( x = 4 - \sqrt{5} \).
5. <Step 5: Verify the solutions.> Substitute each solution back into the original equation to verify they satisfy the equation.>

Key Concepts

Square Root Property

The square root property states that if an equation is in the form of x^2 = k, then the solutions for x can be found by taking the square root of both sides, resulting in x = ±√k. This property is essential for solving quadratic equations, as it allows us to isolate the variable and find its possible values.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. They can be solved using various methods, including factoring, completing the square, and applying the square root property, which is particularly useful when the equation can be expressed in a perfect square form.

Isolating the Variable

Isolating the variable involves rearranging an equation to get the variable on one side and all other terms on the opposite side. This step is crucial in solving equations, as it simplifies the problem and allows for the application of properties like the square root property, making it easier to find the solution.