Solve: -x^4 + 200 = 102x^2
-x^4-102x^2+200=0
(-x^2)^2-102(x^2)+200=0
(x^2-51)(x^2+51)(x-20)(x-10)
I'm stuck at this point
Your last line makes absolutely no sense.
Just multiply the first terms in each bracket all together, you would get x^6 !!! and your last term would be -520200 !!!
Also I think you have a type.
If your equation is
+x^4 + 200 = 102x^2 it works out very nicely, so I am going to assume that the -x^4 is incorrect.
x^4 - 102x^2 + 200 = 0
(x^2 - 2)(x^2 - 100) = 0
so x = ±√2 or x = ±10
To continue solving the equation -x^4 + 200 = 102x^2, you correctly factored the left side as (x^2-51)(x^2+51). However, the right side of the equation is not yet factored completely.
Let's expand the right side and get it into a form that we can factor further:
102x^2 = 2 * 51 * x^2 = 2 * (x^2 - 51)
Now we have the equation in the form:
(x^2-51)(x^2+51) = 2 * (x^2 - 51)
At this point, we can simplify the equation by canceling out the common factor of x^2 - 51 from both sides:
(x^2-51)(x^2+51) - 2 * (x^2 - 51) = 0
Now, let's distribute the 2 to the terms inside the brackets:
(x^2-51)(x^2+51) - 2x^2 + 102 = 0
Expanding the brackets:
x^4 - 51x^2 + 51x^2 - 2601 - 2x^2 + 102 = 0
Combining like terms:
x^4 - x^2 - 2499 = 0
Now the equation becomes:
x^4 - x^2 - 2499 = 0
However, this equation cannot be factored further using the standard factoring techniques. To find the solutions, you will need to use numerical methods such as graphing, factoring, or using a calculator.
If you need help finding the solutions numerically or graphically, please let me know!
To solve the equation -x^4 + 200 = 102x^2, we can start by rewriting it as a quadratic equation by moving all the terms to one side:
-x^4 - 102x^2 + 200 = 0
Next, we can factor out common terms and rewrite the equation as a quadratic expression:
(-x^2)^2 - 102(x^2) + 200 = 0
Simplifying further, we have:
(x^2 - 51)(x^2 + 51) = 0
Now, we have a product of two factors equal to zero. For the equation to be true, either the first factor (x^2 - 51) must be equal to zero, or the second factor (x^2 + 51) must be equal to zero.
So we have two separate equations to solve:
x^2 - 51 = 0
x^2 + 51 = 0
Taking the square root of both sides, we get:
x = ±√51
Similarly:
x = ±√(-51)
However, the √(-51) is not a real number since the square root of a negative number is not defined in the real number system. Therefore, we only have two real solutions:
x = ±√51
So the solutions to the equation -x^4 + 200 = 102x^2 are x = √51 and x = -√51.