Here is a doozy:
Solve: ((-2w^2+v)^2)+4t^4-(-1+v-2t^2)^2
what do you mean by "solve"? It's not an equation.
You can expand it out, but the only thing that drops out are the 4t^4 and v^2 terms.
It doesn't factor to anything simple.
Oh sorry, meant simplify. It's a super strange one I guess but that's what they've given me and I was fairly confused.
( - 2 w² + v )² + 4 t⁴ - ( - 1 + v - 2 t² )² =
( - 2 w² )² + 2 ∙ ( - 2 w ) ∙ v + v ² + 4 t⁴ - ( - 1 + v - 2 t² ) ∙ ( - 1 + v - 2 t² ) =
4 w⁴ - 4 w ∙ v + v ² + 4 t⁴ - [ - 1 ∙ ( - 1 ) + ( - 1 ) ∙ v - ( - 1 ) ∙ ( 2 t² ) + v ∙ ( - 1 ) + v ∙ v + v ∙ ( - 2 t² ) - ( - 2 t² ) ∙ 1 + ( - 2 t² ) ∙ v - ( - 2 t² ) ∙ 2 t² ] =
4 w⁴ - 4 w ∙ v + v ² + 4 t⁴ - ( 1 - v + 2 t² - v + v² - 2 t² ∙ v + 2 t² - 2 t² ∙ v + 4 t⁴ ) =
4 w⁴ - 4 w ∙ v + v ² + 4 t⁴ - 1 + v - 2 t² + v - v² + 2 t² ∙ v - 2 t² + 2 t² ∙ v - 4 t⁴ =
4 w⁴ - 4 w ∙ v + 2 v - 4 t² + 2 t² ∙ v + 2 t² ∙ v - 1 =
2 ( 2 w⁴ - 2 w ∙ v + v - 2 t² + t² ∙ v + t² ∙ v ) - 1
To solve the given expression, ((-2w^2 + v)^2) + 4t^4 - (-1 + v - 2t^2)^2, we'll need to simplify and expand it step by step. Let's begin:
Step 1: Simplify the Expression
The first step is to simplify the given expression by removing any parentheses. We can simplify the expression as follows:
((-2w^2 + v)^2) + 4t^4 - (-1 + v - 2t^2)^2
= (-2w^2 + v) * (-2w^2 + v) + 4t^4 - (-1 + v - 2t^2) * (-1 + v - 2t^2)
Step 2: Expand the Squares
Next, we'll expand the squares using the FOIL (First, Outer, Inner, Last) method. Expanding the squares gives us:
((-2w^2 * -2w^2) + (-2w^2 * v) + (v * -2w^2) + (v * v)) + 4t^4 - ((-1 * -1) + (-1 * v) + (v * -1) + (-1 * -2t^2)) + ((v * v) + (v * -2t^2) + (-2t^2 * v) + (-2t^2 * -2t^2))
Step 3: Simplify the Expression Further
Now, simplify the expression further by multiplying and combining like terms:
(4w^4 - 4w^2v + 4w^2v + v^2) + 4t^4 - (1 + (-v) + (-v) + 2t^2) + (v^2 - 2vt^2 - 2vt^2 + 4t^4)
After simplifying, we have:
4w^4 + v^2 + 4t^4 - 1 - 2v - 4t^2 + v^2 - 4vt^2 + 4t^4
Step 4: Combine Like Terms
Combine the like terms by adding or subtracting them together:
4w^4 + 2v^2 + 8t^4 - 4vt^2 - 4t^2 - 1 - 2v
And that is the final simplified form of the given expression: 4w^4 + 2v^2 + 8t^4 - 4vt^2 - 4t^2 - 1 - 2v.