What are the solutions of the equation [tex]$x^4+95 x^2-500=0$[/tex]? Use factoring to solve.
A. [tex]$x= \pm \sqrt{5}$[/tex] and [tex]$x= \pm 10$[/tex]
B. [tex]$x= \pm i \sqrt{5}$[/tex] and [tex]$x= \pm 10 i$[/tex]
C. [tex]$x= \pm \sqrt{5}$[/tex] and [tex]$x= \pm 10 i$[/tex]
D. [tex]$x= \pm i \sqrt{5}$[/tex] and [tex]$x= \pm 10$[/tex]
Answer (2)
Substitute y = x 2 to transform the equation into y 2 + 95 y − 500 = 0 .
Factor the quadratic equation as ( y + 100 ) ( y − 5 ) = 0 , yielding y = − 100 or y = 5 .
Substitute back x 2 for y , giving x 2 = − 100 or x 2 = 5 .
Solve for x to find the solutions: x = ± 5 and x = ± 10 i . The final answer is x = ± 5 and x = ± 10 i .
Explanation
Problem Analysis We are given the equation x 4 + 95 x 2 − 500 = 0 and asked to find its solutions by factoring.
Substitution Let's make a substitution to simplify the equation. Let y = x 2 . This transforms the equation into a quadratic equation in terms of y : y 2 + 95 y − 500 = 0 .
Factoring Now, we need to factor the quadratic equation y 2 + 95 y − 500 = 0 . We are looking for two numbers that multiply to -500 and add up to 95. These numbers are 100 and -5. So, we can write the quadratic equation in factored form as ( y + 100 ) ( y − 5 ) = 0 .
Solving for y Next, we solve for y . Setting each factor equal to zero gives us: y + 100 = 0 or y − 5 = 0 . Solving these equations, we find y = − 100 or y = 5 .
Substituting Back Now we substitute back x 2 for y to find the values of x . We have two cases: x 2 = − 100 and x 2 = 5 .
Solving for x (Imaginary Roots) For x 2 = − 100 , we take the square root of both sides: x = ± − 100 = ± 10 i . Here, i is the imaginary unit, where i 2 = − 1 .
Solving for x (Real Roots) For x 2 = 5 , we take the square root of both sides: x = ± 5 .
Final Solutions Therefore, the solutions to the equation x 4 + 95 x 2 − 500 = 0 are x = ± 5 and x = ± 10 i .
Examples
Imagine you are designing a suspension bridge and need to calculate the tension in the cables. The equation x 4 + 95 x 2 − 500 = 0 might arise when modeling the forces and deflections in the structure. Solving this equation helps engineers determine critical parameters for ensuring the bridge's stability and safety. Factoring and finding the roots allows for precise calculations of these forces, preventing potential structural failures. This ensures the bridge can withstand various loads and environmental conditions, making it safe for public use.
The solutions to the equation x 4 + 95 x 2 − 500 = 0 are x = ± 5 and x = ± 10 i . Thus, the correct answer is option C. This was achieved by substituting y = x 2 and factoring the resulting quadratic equation.
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