Select the correct answer.
What are the solutions of this quadratic equation?
[tex]x^2+10=0[/tex]
A. [tex]z= \pm \sqrt{10}[/tex]
B. [tex]z= \pm 5[/tex]
C. [tex]z= \pm \sqrt{10} i[/tex]
D. [tex]z= \pm 5 i[/tex]
Answer (2)
To solve the equation x 2 + 10 = 0 , we find that the solutions are x = ± 10 i , indicating that they are complex numbers. The correct option is C: z = ± 10 i .
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Rewrite the equation as x 2 = − 10 .
Take the square root of both sides: x = ± − 10 .
Express the square root of a negative number using the imaginary unit i : x = ± 10 i .
The solutions are z = ± 10 i , which corresponds to option C. z = ± 10 i
Explanation
Understanding the Problem We are given the quadratic equation x 2 + 10 = 0 . Our goal is to find the solutions for x . The variable in the answer options is z , so we can assume z = x .
Isolating the x^2 term First, we rewrite the equation as x 2 = − 10 . This isolates the x 2 term on one side of the equation.
Taking the Square Root Next, we take the square root of both sides of the equation: x = ± − 10 . Remember that when taking the square root, we must consider both the positive and negative roots.
Introducing the Imaginary Unit Since we have a negative number under the square root, we need to use the imaginary unit i , where i = − 1 . Thus, we can rewrite − 10 as 10 ⋅ − 1 = 10 i .
Final Solutions Therefore, the solutions are x = ± 10 i . Comparing this to the given options, we see that option C matches our solution.
Examples
Quadratic equations with complex roots can model oscillating systems in physics, such as the motion of a pendulum with damping or the behavior of an electrical circuit with inductance and capacitance. The imaginary part of the solution indicates the oscillatory nature of the system, while the real part (if any) describes the damping or decay of the oscillations. Understanding complex roots helps engineers design and analyze systems that exhibit oscillatory behavior.
