Question

solve the equation by making an appropriate substitution.
$x^4 - 29x^2 + 100 = 0$
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is { }
(type an integer or a simplified fraction. use a comma to separate answers as needed.)
b. the solution set is $\varnothing$.

Explanation:

Step1: Substitute \( y = x^2 \)

Let \( y = x^2 \), then the equation \( x^4 - 29x^2 + 100 = 0 \) becomes \( y^2 - 29y + 100 = 0 \).

Step2: Factor the quadratic equation

Factor \( y^2 - 29y + 100 \). We need two numbers that multiply to 100 and add to -29. The numbers are -25 and -4. So, \( y^2 - 29y + 100=(y - 25)(y - 4)=0 \).

Step3: Solve for y

Set each factor equal to zero:

• \( y - 25 = 0 \) gives \( y = 25 \)
• \( y - 4 = 0 \) gives \( y = 4 \)

Step4: Substitute back \( y = x^2 \) and solve for x

• For \( y = 25 \), \( x^2 = 25 \), so \( x=\pm5 \)
• For \( y = 4 \), \( x^2 = 4 \), so \( x=\pm2 \)

Answer:

The solution set is \(\{-5, -2, 2, 5\}\)