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

검증된 단계별 안내

Start by substituting the complex number $x = -3 + 4i$ into the quadratic equation $x^2 + 6x + 25 = 0$.

Calculate $x^2$ by squaring $-3 + 4i$. Use the formula $(a + bi)^2 = a^2 + 2abi + (bi)^2$ where $a = -3$ and $b = 4$.

Compute \$6x$ by multiplying 6 with $-3 + 4i$.

Add the results from $x^2$, \$6x$, and the constant term 25 together.

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.