x^4 + 6x^2 = -8 Solve the equation.
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Place all terms to left side of equation:
x^4 + 6x^2 = -8
x^4 + 6x^2 + 8 = 0
This is factorable. But how do you factor this?
What we'll do is list all the factor pairs of 8 and from these, we choose the pair which has a sum of 6.
1 x 8 | 2 x 4 | -1 x -8 | -2 x -4
Here, we can see that the pair with sum of 6 is 2 x 4.
We rewrite the quadratic equation as such:
(x^2 + 2)(x^2 + 4) = 0
We can equate each factor to zero:
(x^2 + 2) = 0
x^2 = -2
x = sqrt(-2)
(x^2 + 4) = 0
x^2 = -4
x = 2*sqrt(-1)
To solve the equation x^4 + 6x^2 = -8, we can follow these steps:
1. Begin by moving all the terms to one side of the equation to obtain a quadratic equation: x^4 + 6x^2 + 8 = 0.
2. Let's introduce a substitution to simplify the equation. We'll let y = x^2. With this substitution, the equation becomes y^2 + 6y + 8 = 0.
3. Now we have a quadratic equation in terms of y. We can solve it by factoring or using the quadratic formula. In this case, the quadratic equation factors as (y + 2)(y + 4) = 0.
4. Set each factor equal to zero:
y + 2 = 0 or y + 4 = 0.
5. Solve for y in each equation:
For y + 2 = 0, we subtract 2 from both sides to get y = -2.
For y + 4 = 0, we subtract 4 from both sides to get y = -4.
6. Remember that y = x^2, so substitute back in these values to find the possible values for x:
If y = -2, then x^2 = -2, which has no real solutions since the square of a real number is always non-negative.
If y = -4, then x^2 = -4, which also has no real solutions for the same reason.
7. Therefore, the equation x^4 + 6x^2 = -8 has no real solutions.