Solve each equation using the square root property. See Example 2. (-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 $b=-8$.
2. The square root property states that if $a^2=b$, then $a=\pm \sqrt{b}$.
3. Apply the square root property: $-2x+5=\pm \sqrt{-8}$.
4. Simplify $\sqrt{-8}$ by expressing it as $\sqrt{8}\cdot i$, where $i$ is the imaginary unit, and $\sqrt{8}=2\sqrt{2}$.
5. Solve for $x$ by isolating it: $-2x=-5\pm 2\sqrt{2}i$, then divide by -2.

Key Concepts

Square Root Property

The square root property states that if a squared expression equals a number, then the original expression can be solved by taking the square root of both sides. This property is particularly useful for equations in the form (ax + b)^2 = c, allowing us to isolate the variable by applying the square root to both sides, considering both the positive and negative roots.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'i' is the imaginary unit defined as the square root of -1. In the context of the given equation, if the right side of the equation is negative, the solutions will involve complex numbers, indicating that the original equation has no real solutions.

Isolating the Variable

Isolating the variable is a fundamental algebraic technique used to solve equations. It involves rearranging the equation to get the variable on one side and all other terms on the opposite side. In the context of the given equation, this means simplifying the expression before applying the square root property, ensuring that the variable is clearly defined for solving.