Question

Find the real or imaginary solutions of the equations by factoring. x^4-26x^2=-25

Answer 1

x=1, -1, 5, and -5.

Explanation:

Given the equation:

`x^4-26x^2=-25`

First, bring the constant term to the left-hand side of the equation:

`x^4-26x^2+25=0`

Next, rewrite the middle term using factors of 25x⁴.

`x^4-25x^2-x^2+25=0`

Factor the first two and last two terms:

`\begin{gathered} x^2(x^2-25)-1(x^2-25)=0 \\ (x^2-1)(x^2-25)=0 \\ (x^2-1)(x^2-5^2)=0 \end{gathered}`

Using the principle of difference of two squares: a²-b²=(a-b)(a+b)

The solutions are 1, -1, 5, and -5.