Apply the square root principle to solve $(x-3)^2+9=0$.
A. $x=-3+3 i,-3-3 i$
B. $x=0,6$
C. $x=3+3 i, 3-3 i$
D. $x=0,-6$
Answer (1)
Isolate the squared term: ( x − 3 ) 2 = − 9 .
Apply the square root principle: x − 3 = ± − 9 .
Simplify the square root: x − 3 = ± 3 i .
Solve for x : x = 3 ± 3 i , so x = 3 + 3 i and x = 3 − 3 i . The answer is x = 3 + 3 i , 3 − 3 i .
Explanation
Understanding the Problem We are given the equation ( x − 3 ) 2 + 9 = 0 and asked to solve for x using the square root principle. This principle states that if u 2 = a , then u = ± a .
Isolating the Squared Term First, we isolate the squared term by subtracting 9 from both sides of the equation: ( x − 3 ) 2 = − 9
Applying the Square Root Principle Next, we apply the square root principle: x − 3 = ± − 9
Simplifying the Square Root Now, we simplify the square root. Since − 1 = i , we have − 9 = 9 × − 1 = 9 × − 1 = 3 i . Therefore, x − 3 = ± 3 i
Solving for x Finally, we solve for x by adding 3 to both sides of the equation: x = 3 ± 3 i
This gives us two solutions: x = 3 + 3 i and x = 3 − 3 i .
Final Answer The solutions to the equation ( x − 3 ) 2 + 9 = 0 are x = 3 + 3 i and x = 3 − 3 i .
Examples
Complex numbers might seem abstract, but they're incredibly useful in fields like electrical engineering. Imagine designing a circuit where you need to analyze alternating current (AC). AC voltage and current can be represented using complex numbers, where the real part represents the resistance and the imaginary part represents the reactance (opposition to current due to capacitance and inductance). By solving equations with complex numbers, engineers can predict the behavior of the circuit and optimize its performance. This ensures that your electronic devices work efficiently and safely!
