Use the equation y=x4−2x2b2+b4 to answer the question.
How can you rewrite the given formula to correctly isolate x?
y = x^4 - 2x^2 b^2 + b^4
y = (x^2 - b^2)^2
x^2 - b^2 = ±√y
x^2 = b^2±√y
x = ±√(b^2±√y)
To rewrite the formula to isolate x, we need to rearrange the equation so that x is on one side of the equation by itself. Here's the step-by-step process:
1. Start with the equation: y = x^4 - 2x^2b^2 + b^4
2. Subtract y from both sides of the equation: y - y = x^4 - 2x^2b^2 + b^4 - y
3. Rearrange the terms so that x terms are on one side: x^4 - 2x^2b^2 = y - b^4
4. Factor out the common factor of x^2: x^2(x^2 - 2b^2) = y - b^4
5. Divide both sides by (x^2 - 2b^2): x^2 = (y - b^4) / (x^2 - 2b^2)
6. Take the square root of both sides to isolate x: x = ±√((y - b^4) / (x^2 - 2b^2))
Note that when you take the square root, you get both positive and negative solutions for x, so there are two possible values.