Question

solve the equation by making an appropriate substitution.\\(x^{4}-25x^{2}+144 = 0\\)\\(\\)\\(\\)select the correct choice below and, if necessary, fill in the answer box to complete your choice\\(\\)\\(\\)\\(\circ\\) a. the solution set is \\(\\{\\}\\)\\(\\)\\(\\) (type an integer or a simplified fraction. use a comma to separate answers as needed.)\\(\\)\\(\circ\\) b. the solution set is \\(\varnothing\\)

Explanation:

Step1: Let \( u = x^2 \)

Substitute \( u \) into the equation, we get \( u^2 - 25u + 144 = 0 \)

Step2: Factor the quadratic equation

We need to find two numbers that multiply to 144 and add up to -25. The numbers are -9 and -16. So, \( (u - 9)(u - 16) = 0 \)

Step3: Solve for \( u \)

Set each factor equal to zero:

• \( u - 9 = 0 \) gives \( u = 9 \)
• \( u - 16 = 0 \) gives \( u = 16 \)

Step4: Substitute back \( u = x^2 \)

• For \( u = 9 \), \( x^2 = 9 \), so \( x = \pm 3 \)
• For \( u = 16 \), \( x^2 = 16 \), so \( x = \pm 4 \)

Answer:

-3, 3, -4, 4