2x^2-10x+25-x^2=0,x^4-1/x^4
You asked:
Given \(2 \cdot {x}^{2} - 10 x + 25 - {x}^{2} = 0\) evaluate the expression \({x}^{4} - \frac{1}{{x}^{4}}\).
MathBot Answer:
\[x^{4} - \frac{1}{x^{4}} = \frac{390624}{625}\]
Solve by factoring:
Factor the nonzero side of the equation completely. $$\begin{aligned} - x^{2} + 2 x^{2} - 10 x + 25 &= 0 \\ \left(x - 5\right)^{2} &= 0\end{aligned}$$
The product of any number and zero is zero, so the equation will hold true if any factors of the quadratic are zero. Set the factors equal to zero and solve. Since the quadratic is square, there is only one factor to set equal to zero and only one solution. $$\begin{aligned} x - 5 &= 0\\ x &= 5\end{aligned}$$
By substituting this value into the expression, we find that: $$\begin{aligned}x^{4} - \frac{1}{x^{4}} &= - \frac{1}{5^{4}} + 5^{4}\\&= \frac{390624}{625}\end{aligned}$$