Given $2 x^2=-8, x$ is
Answer (2)
Divide both sides of the equation by 2: x 2 = − 4 .
Take the square root of both sides: x = ± − 4 .
Simplify the square root using the imaginary unit i : x = ± 2 i .
The solutions are x = 2 i and x = − 2 i , so the final answer is 2 i , − 2 i .
Explanation
Problem Analysis We are given the equation 2 x 2 = − 8 and asked to find the value of x .
Isolating x 2 First, we want to isolate x 2 by dividing both sides of the equation by 2: 2 2 x 2 = 2 − 8 x 2 = − 4
Taking the Square Root Now, we take the square root of both sides of the equation to solve for x :
x = ± − 4
Simplifying the Square Root Since we have a negative number under the square root, we know that the solutions will be complex numbers. Recall that − 1 = i , 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, like the ones we found as solutions to this equation, are used in electrical engineering to analyze alternating current circuits. They help in representing quantities that have both magnitude and phase, such as voltage and current in AC circuits. Understanding complex numbers is crucial for designing and analyzing various electronic systems.
The solutions to the equation 2 x 2 = − 8 are complex numbers, specifically x = 2 i and x = − 2 i . This is achieved by isolating x 2 , taking the square root, and simplifying using the imaginary unit i .
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