Solve: -x^4 + 200 = 102x^2
-x^4-102x²+200=0
X=x²
-X²-102X+200=0
try to solve this equation and then u can solve -x^4-102x²+200=0
-x^4-102x^2+200=0
(-x^2)^2-102(x^2)+200=0
I'm stuck at this point
To solve the equation -x^4 + 200 = 102x^2, we can rearrange it to set it equal to zero and then try to factor it. Here's how:
1. Start with the equation: -x^4 + 200 = 102x^2
2. Move all the terms to the left side to set the equation equal to zero: -x^4 - 102x^2 + 200 = 0
3. Next, let's try to factor the equation. Notice that all the terms have a common factor of -1, so we can factor that out: -1(x^4 + 102x^2 - 200) = 0
4. Now, let's focus on factoring the expression inside the parentheses: x^4 + 102x^2 - 200. We can see that this is a quadratic expression in terms of x^2.
5. To factor the quadratic expression, let's use the fact that the sum of the roots and the product of the roots can help us determine the factorization. Let's assume the roots of the quadratic expression are a and b.
6. The sum of the roots (a + b) is given by -b1/a1, where a1 and b1 are the coefficients of x^2 and the constant term, respectively. In this case, a1 = 1 and b1 = -200, so the sum of the roots is -(102/1) = -102.
7. The product of the roots (ab) is given by c1/a1, where c1 is the constant term and a1 is the coefficient of x^2. In this case, c1 = -200 and a1 = 1, so the product of the roots is (-200/1) = -200.
8. Now, we need to find the pair of numbers that add up to -102 and multiply to -200. Through trial and error, we find that the pair is -2 and -100.
Now, we can use these factors to write our equation as a product of linear factors:
-1(x^4 + 102x^2 - 200) = 0
-1(x^2 - 2)(x^2 - 100) = 0
9. Finally, to find the values of x, we set each factor equal to zero and solve for x:
x^2 - 2 = 0 --> x^2 = 2 --> x = ±√2
x^2 - 100 = 0 --> x^2 = 100 --> x = ±10
So, the solutions to the equation -x^4 + 200 = 102x^2 are x = -√2, √2, -10, 10.