I have a problem that states to simplify.
The equation is d=sqrt(x^2+(4-x^2-2)^2)
The book tells me that the answer simplifies to sqrt(x^4-3x^2+4).
Can you please help me understand where the -3x^2 came from?
Thanks!
d=√(x^2 + (4 - x^2 - 2)^2)
=√(x^2 + (2 - x^2)^2)
=√(x^2 + x^4 - 4x^2 + 4)
=√(x^4 - 3x^2 + 4)
In my opinion, this is not much of a "simplification" problem
(4-x^2-2)^2 = (2 - x^2)^2 = x^4 - 4x^2 + 4
When you add the x^2 to that, you get
x^4 - 3x^2 + 4
(4 - x^2 - 2)^2 simplifies to (2 - x^2)^2. Expanding it gives x^4 - 2x^2 - 2x^2 + 4 = x^4 - 4x^4 + 4. Then you have that x^2 out in front that you can work with. so x^4 + x^2 - 4x^2 + 4 = x^4 - 3x^2 + 4.
To clarify further, let's break down the expression step by step:
1. Start with the equation: d = √(x^2 + (4 - x^2 - 2)^2)
2. Simplify the expression within the square root:
(4 - x^2 - 2)^2 = (2 - x^2)^2
3. Expand (2 - x^2)^2 using the FOIL method:
(2 - x^2)^2 = (2 - x^2)(2 - x^2)
= 2(2) + 2(-x^2) + (-x^2)(2) + (-x^2)(-x^2)
= 4 - 2x^2 - 2x^2 + x^4
= x^4 - 4x^2 + 4
4. Substitute the expanded expression back into the original equation:
d = √(x^2 + (x^4 - 4x^2 + 4))
5. Simplify further by combining like terms:
d = √(x^4 - 3x^2 + 4)
Therefore, the simplified expression is √(x^4 - 3x^2 + 4), which is where the -3x^2 term came from.