Find the real or imaginary solutions of the equation by factoring.
x^4-20x^2=-64
To begin factoring the equation, let's first rearrange it to have all terms on one side:
x^4 - 20x^2 + 64 = 0
Now, we can see that this is a quadratic equation in terms of x^2. Let's assign a variable to make it clearer:
y = x^2
Now, we can rewrite the equation in terms of y:
y^2 - 20y + 64 = 0
To factor this quadratic equation, we need to find two numbers that multiply to 64 and add up to -20. These numbers are -4 and -16. We can rewrite the equation using these numbers:
(y - 4)(y - 16) = 0
Now, let's replace y with x^2:
(x^2 - 4)(x^2 - 16) = 0
To find the solutions, we can solve each factor separately:
x^2 - 4 = 0
(x + 2)(x - 2) = 0
x + 2 = 0 or x - 2 = 0
x = -2 or x = 2
x^2 - 16 = 0
(x + 4)(x - 4) = 0
x + 4 = 0 or x - 4 = 0
x = -4 or x = 4
Therefore, the solutions to the equation x^4 - 20x^2 + 64 = 0 are x = -2, x = 2, x = -4, and x = 4.
To solve the equation x^4 - 20x^2 = -64 by factoring, we can start by moving all the terms to one side to get:
x^4 - 20x^2 + 64 = 0
Now, let's try to factor the equation. The expression x^4 - 20x^2 + 64 can be factored as a perfect square trinomial:
(x^2 - 8)^2 = 0
Now, let's solve for x using the zero product property:
x^2 - 8 = 0
By setting each factor equal to zero, we can solve for x:
x^2 - 8 = 0
Adding 8 to both sides:
x^2 = 8
Taking the square root of both sides:
x = ± √8
Simplifying the square root of 8:
x = ± 2√2
Therefore, the real solutions to the equation x^4 - 20x^2 = -64 are x = 2√2 and x = -2√2.