Work each problem. Show that -2+i is a solution of the equation x²+4x+5=0.

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Start by understanding that to show $-2 + i$ is a solution of the equation $x^2 + 4x + 5 = 0$, you need to substitute $x = -2 + i$ into the equation and verify that the left-hand side equals zero.

Substitute $x = -2 + i$ into the equation: calculate $(-2 + i)^2 + 4(-2 + i) + 5$.

First, expand $(-2 + i)^2$ using the formula $(a + b)^2 = a^2 + 2ab + b^2$: \[(-2)^2 + 2(-2)(i) + i^2\]

Next, simplify the terms: - $(-2)^2 = 4$ - $2(-2)(i) = -4i$ - $i^2 = -1$ (since $i$ is the imaginary unit) So, $(-2 + i)^2 = 4 - 4i - 1 = 3 - 4i$.

Now, substitute back and simplify the entire expression: \[(3 - 4i) + 4(-2 + i) + 5\] Calculate $4(-2 + i) = -8 + 4i$, then combine all terms: \[3 - 4i - 8 + 4i + 5\] Group like terms and simplify to check if the result equals zero.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Numbers

Complex numbers include a real part and an imaginary part, expressed as a + bi, where i is the imaginary unit with the property i² = -1. Understanding how to work with complex numbers is essential for evaluating expressions and verifying solutions involving imaginary components.

Substitution Method

The substitution method involves replacing the variable in an equation with a given value to check if it satisfies the equation. Here, substituting x = -2 + i into the quadratic equation helps verify whether it is a root.

Quadratic Equations and Roots

Quadratic equations are polynomial equations of degree two, typically in the form ax² + bx + c = 0. Their solutions, or roots, can be real or complex, and verifying a root means confirming that substituting it into the equation yields zero.

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