Question

Solve the equation by identifying the quadratic form. Use a substitute variable(t) and find all real solutions by factoring. Type your answers from smallest to largest. If an answer is not an integer then type it as a decimal rounded to the nearest hundredth. When typing exponents do not use spaces and use the carrot key ^ (press shift and 6). For example, x cubed can be typed as x^3.x^{10}-2x^5+1=0Step 1. Identify the quadratic formLet t= Answer. We now have:t^2-2t+1=0Step 2. FactorFactor this and solve for t to get t=Answer Step 3. Solve for xWe have solved for t now we need to use this value for t to help us solve for x. Revisit step 1 to remind you of the relationship between t and x. Type your real solutions (no extraneous) from smallest to largest.x= Answer

Answer 1

Given:

`x^(10)-2x^5+1=0`

Step 1: To identify the quadratic form of the given equation.

`\begin{gathered} x^(10)-2x^5+1=0 \\ (x^5)^2-2x^5+1=0 \\ \text{Put x}^5=t,\text{ it gives} \\ t^2-2t+1=0 \end{gathered}`

So, t = x²

Step 2: Factor the quadratic equation in step 1.

`\begin{gathered} t^2-2t+1=0 \\ t^2-t-t+1=0 \\ t(t-1)-t(t-1)=0 \\ (t-1)(t-1)=0 \end{gathered}`

Thus, the factors of the equation is

`(t-1)(t-1)=0`

Step3: solve for x.

`\begin{gathered} (t-1)(t-1)=0 \\ (x^5-1)(x^5-1)=0 \\ \Rightarrow x^5-1=0,x^5-1=0 \\ \Rightarrow x=1 \end{gathered}`

Answer: x = 1