Solve: x^4 - 2x^2 = 35
To solve the equation x^4 - 2x^2 = 35, we can start by rearranging the equation to have all terms on one side:
x^4 - 2x^2 - 35 = 0
Now, we can notice that the equation is a quadratic equation in terms of x^2. Let's substitute y = x^2 to simplify the equation:
y^2 - 2y - 35 = 0
Next, we can factor the quadratic equation:
(y - 7)(y + 5) = 0
Setting each factor equal to zero and solving for y:
y - 7 = 0 or y + 5 = 0
y = 7 or y = -5
Now, substitute back x^2 for y in each equation:
x^2 = 7 or x^2 = -5
To solve for x, take the square root of both sides:
x = ±√7 or x = ±√(-5)
Since we cannot take the square root of a negative number in the real number system, the solutions are:
x = ±√7
So the equation x^4 - 2x^2 = 35 has two real solutions: x = √7 and x = -√7.