What are the solutions of the equation [tex]$9 x^4-2 x^2-7=0$[/tex]? Use [tex]$u$[/tex] substitution to solve.
A. [tex]$x= \pm \sqrt{\frac{7}{9}}$[/tex] and [tex]$x= \pm 1$[/tex]
B. [tex]$x= \pm \sqrt{\frac{7}{9}}$[/tex] and [tex]$x= \pm i$[/tex]
C. [tex]$x= \pm i \sqrt{\frac{7}{9}}$[/tex] and [tex]$x= \pm 1$[/tex]
D. [tex]$x= \pm i \sqrt{\frac{7}{9}}$[/tex] and [tex]$x= \pm i$[/tex]
Answer (2)
The solutions to the equation 9 x 4 − 2 x 2 − 7 = 0 using u substitution lead to x = ± 1 and x = ± i 9 7 . Therefore, the correct answer is option C.
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Substitute u = x 2 to transform the equation into a quadratic equation: 9 u 2 − 2 u − 7 = 0 .
Solve the quadratic equation for u using the quadratic formula, obtaining u = 1 and u = − 9 7 .
Substitute back x 2 for u , giving x 2 = 1 and x 2 = − 9 7 .
Solve for x , resulting in the solutions x = ± 1 and x = ± i 9 7 . The final answer is x = ± 1 , ± i 9 7
Explanation
Understanding the Problem We are given the equation 9 x 4 − 2 x 2 − 7 = 0 and asked to solve for x using u substitution. This means we will replace x 2 with a new variable, u , to transform the equation into a quadratic equation, which we can then solve more easily. After finding the values of u , we will substitute back x 2 for u and solve for x .
Making the Substitution Let u = x 2 . Substituting this into the given equation, we get: 9 u 2 − 2 u − 7 = 0
Applying the Quadratic Formula Now we solve the quadratic equation 9 u 2 − 2 u − 7 = 0 for u . We can use the quadratic formula: u = 2 a − b ± b 2 − 4 a c where a = 9 , b = − 2 , and c = − 7 .
Calculating the Values of u First, calculate the discriminant: D = b 2 − 4 a c = ( − 2 ) 2 − 4 ( 9 ) ( − 7 ) = 4 + 252 = 256 Now, find the values of u :
u = 18 2 ± 256 = 18 2 ± 16 So, u 1 = 18 2 + 16 = 18 18 = 1 and u 2 = 18 2 − 16 = 18 − 14 = − 9 7 .
Substituting Back Since u = x 2 , we have x 2 = 1 and x 2 = − 9 7 . Now we solve for x in each case.
Solving for x For x 2 = 1 , we have x = ± 1 = ± 1 .
For x 2 = − 9 7 , we have x = ± − 9 7 = ± i 9 7 .
Final Answer Therefore, the solutions are x = ± 1 and x = ± i 9 7 .
Conclusion The solutions to the equation 9 x 4 − 2 x 2 − 7 = 0 are x = ± 1 and x = ± i 9 7 .
Examples
In electrical engineering, solving equations of this form can help determine the impedance in AC circuits. The roots of the equation represent the frequencies at which the circuit will resonate, which is crucial for designing filters and amplifiers. Understanding these resonant frequencies ensures that the circuit operates efficiently and avoids unwanted signal amplification. By finding the values of x, engineers can optimize circuit performance and prevent potential damage from excessive resonance.
