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

Find The Real Or Imaginary Solutions Of The Equations By Factoring X426x225 class=

Sagot :

Answer:

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

Explanation:

Given the equation:

[tex]x^4-26x^2=-25[/tex]

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

[tex]x^4-26x^2+25=0[/tex]

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

[tex]x^4-25x^2-x^2+25=0[/tex]

Factor the first two and last two terms:

[tex]\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}[/tex]

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

[tex]\begin{gathered} (x-1)(x+1)(x-5)(x+5)=0 \\ \implies x-1=0\text{ or }x+1=0\text{ or }x-5=0\text{ or }x+5=0 \\ \operatorname{\implies}x=1\text{ or }x=-1\text{ or }x=5\text{ or }x=-5 \end{gathered}[/tex]

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

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