Question

solve the equation by making an appropriate substitution.\\(x^{4}-25x^{2}+144 = 0\\)

Explanation:

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

Let \( y = x^2 \), then the equation \( x^4 - 25x^2 + 144 = 0 \) becomes \( y^2 - 25y + 144 = 0 \).

Step2: Factor the quadratic equation

Factor \( y^2 - 25y + 144 = 0 \). We need two numbers that multiply to 144 and add to -25. Those numbers are -9 and -16. So, \( (y - 9)(y - 16) = 0 \).

Step3: Solve for \( y \)

Set each factor equal to zero:

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

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

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

Answer:

\( x = -4, -3, 3, 4 \)