Solve the equation.
$9 x^2+4=0$
A. $\pm \frac{3}{2} i$
B. $\pm \frac{2}{3} i$
C. $\pm \frac{9}{4} i$
D. $\pm \frac{4}{9} i$
Answer (1)
Isolate the x 2 term: 9 x 2 = − 4 .
Divide by 9: x 2 = − 9 4 .
Take the square root: x = } .
Simplify using the imaginary unit: x = } .
Explanation
Understanding the Problem We are given the equation 9 x 2 + 4 = 0 and asked to solve for x . This involves finding the values of x that satisfy the equation.
Isolating the x 2 Term First, we isolate the x 2 term by subtracting 4 from both sides of the equation:
9 x 2 = − 4
Dividing by 9 Next, we divide both sides by 9 to get x 2 by itself:
x 2 = − 9 4
Taking the Square Root Now, we take the square root of both sides. Remember that when taking the square root, we consider both positive and negative roots:
x = } . Since we have a negative number under the square root, we'll use the imaginary unit i , where i = } .
x = } .
Simplifying the Square Root We simplify the square root:
x = } .
x = } .
Final Answer Therefore, the solutions to the equation are x = 3 2 i and x = − 3 2 i . We can write this as x = } .
Examples
Complex numbers might seem abstract, but they're incredibly useful in electrical engineering. Imagine designing a circuit where you need to analyze alternating current (AC). AC voltages and currents can be represented as complex numbers, making calculations much easier. For example, the impedance of a circuit, which is the opposition to the flow of current, is often a complex number. By solving equations with complex numbers, engineers can optimize circuit designs for efficiency and stability.
