Question
Find the real solutions of the equation -27=x^4-12x^2
Answers
Answered by
GPT 3.5
To solve the equation -27 = x^4 - 12x^2, rearrange the equation to get x^4 - 12x^2 + 27 = 0.
Let's substitute y = x^2 to simplify the equation: y^2 - 12y + 27 = 0.
Factor the quadratic equation: (y - 9)(y - 3) = 0.
Set each factor equal to zero and solve for y:
y - 9 = 0 -> y = 9
y - 3 = 0 -> y = 3
Now, substitute back y = x^2 to solve for x:
When y = 9:
x^2 = 9 -> x = ±√9 -> x = ±3
When y = 3:
x^2 = 3 -> x = ±√3
Therefore, the real solutions of the equation -27 = x^4 - 12x^2 are x = -3, x = 3, x = -√3 and x = √3.
Let's substitute y = x^2 to simplify the equation: y^2 - 12y + 27 = 0.
Factor the quadratic equation: (y - 9)(y - 3) = 0.
Set each factor equal to zero and solve for y:
y - 9 = 0 -> y = 9
y - 3 = 0 -> y = 3
Now, substitute back y = x^2 to solve for x:
When y = 9:
x^2 = 9 -> x = ±√9 -> x = ±3
When y = 3:
x^2 = 3 -> x = ±√3
Therefore, the real solutions of the equation -27 = x^4 - 12x^2 are x = -3, x = 3, x = -√3 and x = √3.
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