Work each problem. Show that -3+4i is a solution of the equation x²+6x+25=0.

Verified step by step guidance

1. Start by substituting the complex number \(x = -3 + 4i\) into the quadratic equation \(x^2 + 6x + 25 = 0\).
2. Calculate \(x^2\) by squaring \(-3 + 4i\). Use the formula \((a + bi)^2 = a^2 + 2abi + (bi)^2\) where \(a = -3\) and \(b = 4\).
3. Compute \$6x\( by multiplying 6 with \)-3 + 4i$.
4. Add the results from \(x^2\), \$6x$, and the constant term 25 together.
5. Simplify the expression by combining like terms (real and imaginary parts separately) and verify if the sum equals 0, which confirms that \(-3 + 4i\) is a solution.

Key Concepts

Complex Numbers

Complex numbers are numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit with the property i² = -1. Understanding how to work with complex numbers is essential for evaluating expressions involving imaginary parts.

Substitution in Polynomial Equations

Substitution involves replacing the variable in an equation with a given value to verify if it satisfies the equation. Here, substituting -3 + 4i into the quadratic equation tests whether it is a root.

Simplifying Expressions with Imaginary Numbers

Simplifying expressions with imaginary numbers requires applying algebraic operations and using i² = -1 to reduce terms. This process helps determine if the substituted value makes the equation true.