Question

2 STEP PROBLEM:STEP 1:Completely factor the left-hand side of the equation into two or more factors: 81x^4 - 64x^2 = 0STEP 2:Solve the given equation by factoring. Write your answer in reduced fraction form, if necessary. Separate multiple solutions with a comma. 81x^4 - 64x^2 = 0

Answer 1

We want to factor the equation

$81x^4-64x^2=0$

But this is a difference of 2 squares problem, since

$81x^4=(9x^2)^2$

and

$64x^2=(8x)^2$

Thus,

$\begin{gathered} 81x^4-64x^2=(9x^2)^2-(8x)^2 \\ =(9x^2-8x)(9x^2+8x)=0 \end{gathered}$

This can further be factorized as,

$(x)(x)(9x-8)(9x+8)$

This is the solution to STEP 1, as you can see, there are 4 factors.

2. We want to get the solutions from this factored form, i.e the values of x that makes

$(x)(x)(9x-8)(9x+8)=0$

We see that this happens, when x = 0 twice, and when

$\begin{gathered} 9x-8=0 \\ 9x=8 \\ x=\frac{8}{9} \end{gathered}$

also when,

$\begin{gathered} 9x+8=0 \\ 9x=-8 \\ x=-\frac{8}{9} \end{gathered}$

Therefore the solutions are

$0,0,\frac{8}{9},-\frac{8}{9}$

This is the answer to Part B