Solve each equation using the square root property. (-2x + 5)² = -8

Verified step by step guidance

1. Recognize that the equation is in the form \( (A)^2 = B \), where \( A = -2x + 5 \) and \( B = -8 \). The square root property states that if \( A^2 = B \), then \( A = \pm \sqrt{B} \).
2. Apply the square root property to the equation: \( -2x + 5 = \pm \sqrt{-8} \).
3. Since \( \sqrt{-8} \) involves the square root of a negative number, rewrite it using imaginary numbers: \( \sqrt{-8} = \sqrt{8} \cdot i = 2\sqrt{2}i \). So the equation becomes \( -2x + 5 = \pm 2\sqrt{2}i \).
4. Set up two separate equations to solve for \( x \): 1) \( -2x + 5 = 2\sqrt{2}i \) 2) \( -2x + 5 = -2\sqrt{2}i \).
5. Solve each equation for \( x \) by isolating \( x \): Subtract 5 from both sides, then divide by \( -2 \) to get \( x = \frac{5 \mp 2\sqrt{2}i}{2} \). This gives the two complex solutions.

Key Concepts

Square Root Property

The square root property states that if an equation is in the form (expression)^2 = k, then the solution can be found by taking the square root of both sides, resulting in expression = ±√k. This method is used to solve quadratic equations that are already isolated as a perfect square.

Solving Quadratic Equations

Quadratic equations involve variables raised to the second power. Solving them often requires isolating the squared term and then applying methods like factoring, completing the square, or using the square root property to find the variable's values.

Real and Complex Solutions

When solving equations like (expression)^2 = negative number, the solutions are not real because the square root of a negative number is imaginary. Understanding the difference between real and complex solutions is essential to correctly interpret the results.