Given [tex]$2 x^2=-8$[/tex], x is
Answer (2)
Divide both sides of the equation by 2: x 2 = − 4 .
Take the square root of both sides: x = ± − 4 .
Express the square root in terms of the imaginary unit i : x = ± 2 i .
The solutions are x = ± 2 i .
Explanation
Understanding the Problem We are given the equation 2 x 2 = − 8 and we need to find the value(s) of x that satisfy this equation.
Isolating x 2 First, we divide both sides of the equation by 2 to isolate x 2 :
2 2 x 2 = 2 − 8 x 2 = − 4
Taking the Square Root Next, we take the square root of both sides of the equation: x = ± − 4
Expressing in terms of i Since we have a negative number under the square root, we need to use imaginary numbers. Recall that i = − 1 , so we can rewrite the square root as: x = ± 4 ⋅ − 1 = ± 4 ⋅ − 1 = ± 2 i
Final Answer Therefore, the solutions are x = 2 i and x = − 2 i .
Examples
Complex numbers, which include imaginary numbers like the ones we found in this problem, are used extensively in electrical engineering. For example, they are used to analyze alternating current (AC) circuits. The impedance of a circuit, which is the opposition to the flow of current, is often expressed as a complex number. Understanding how to solve equations involving complex numbers is crucial for designing and analyzing electrical circuits.
To solve 2 x 2 = − 8 , we divide by 2 to get x 2 = − 4 , then take the square root to find x = ± 2 i . The solutions involve imaginary numbers since the square root of a negative number is not defined in real numbers. Thus, x = 2 i and x = − 2 i are the final answers.
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