Solve [tex]2 x^2+26=0[/tex] to identify the roots.
A. [tex]x=-\sqrt{13}, x=\sqrt{13}[/tex]
B. [tex]x=-i \sqrt{13}, x=i \sqrt{13}[/tex]
C. [tex]x=-2 \sqrt{26}, x=2 \sqrt{26}[/tex]
D. [tex]x=-2 i \sqrt{26}, x=2 i \sqrt{26}[/tex]
Answer (1)
Divide the equation by 2: x 2 + 13 = 0 .
Isolate x 2 : x 2 = − 13 .
Take the square root: x = ± − 13 .
Express with imaginary unit: x = ± i 13 .
The roots are x = − i 13 , x = i 13 .
Explanation
Understanding the Problem We are given the equation 2 x 2 + 26 = 0 and asked to find its roots. This means we need to solve for x .
Simplifying the Equation First, let's divide both sides of the equation by 2 to simplify it: 2 2 x 2 + 2 26 = 2 0 x 2 + 13 = 0
Isolating the x^2 Term Next, subtract 13 from both sides of the equation to isolate the x 2 term: x 2 + 13 − 13 = 0 − 13 x 2 = − 13
Taking the Square Root Now, take the square root of both sides of the equation. Remember that when taking the square root, we need to consider both positive and negative roots: x = ± − 13
Expressing Roots with Imaginary Unit Since we have a negative number under the square root, we'll use the imaginary unit i , where i = − 1 . So, we can rewrite the roots as: x = ± 13 × − 1 x = ± 13 × − 1 x = ± i 13
Final Answer Therefore, the roots of the equation 2 x 2 + 26 = 0 are x = − i 13 and x = i 13 .
Examples
Imagine you're designing an electrical circuit and need to determine the impedance, which involves solving quadratic equations that might have complex roots. These roots help you understand the circuit's behavior with alternating current. Similarly, in quantum mechanics, solving the Schrodinger equation for certain potentials can lead to complex solutions, which describe the probability amplitudes of particles in those potentials. Understanding how to solve such equations is crucial for analyzing these systems.
