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The solutions for the equations are i and √2i for the first part, √(1 - √10) and √(1 + √10) for the second part and √((-5 + 3√3) / 2) and √((-5 - 3√3) / 2) for the third part.
To find the solutions of the equation x^4 + 3x^2 + 2 = 0, we can use a method called U-substitution. Here's how it works:
First, we need to identify a substitution that will allow us to simplify the equation. In this case, we can use the substitution u = x^2, which allows us to rewrite the equation as u^2 + 3u + 2 = 0. This is a much simpler equation to solve, as it is a quadratic equation in the variable u.
To solve this equation, we can use the quadratic formula:
u = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = 3, and c = 2, so the quadratic formula becomes:
u = (-3 ± √(3^2 - 4(1)(2))) / 2(1)
Solving for u, we get:
u = (-3 ± √(9 - 8)) / 2
So, the solutions for u are:
u = (-3 + 1) / 2 = -1
u = (-3 - 1) / 2 = -2
Since we made the substitution u = x^2, the solutions for x are the square roots of the solutions for u. So, the solutions for x are:
x = √(-1) = i
x = √(-2) = √2i
Here, i is the imaginary unit, which satisfies the equation i^2 = -1.
Therefore, the solutions of the equation x^4 + 3x^2 + 2 = 0 are x = i and x = √2i.
If you want to find the solutions of the equation x^4 + 6x^2 + 5 = 0, you can use the same method as above. Making the substitution u = x^2, we get the equation u^2 + 6u + 5 = 0. Solving this equation using the quadratic formula, we get:
u = (-6 ± √(6^2 - 4(1)(5))) / 2(1)
Solving for u, we get:
u = (-6 + 2√10) / 2 = 1 - √10
u = (-6 - 2√10) / 2 = 1 + √10
Therefore, the solutions for x are the square roots of the solutions for u:
x = √(1 - √10)
x = √(1 + √10)
To find the solutions of the equation x^4 + 5x^2 + 14 = 0, you can use the same method as above. Making the substitution u = x^2, we get the equation u^2 + 5u + 14 = 0. Solving this equation using the quadratic formula, we get:
u = (-5 ± √(5^2 - 4(1)(14))) / 2(1)
Solving for u, we get:
u = (-5 + 3√3) / 2
u = (-5 - 3√3) / 2
Therefore, the solutions for x are the square roots of the solutions for u:
x = √((-5 + 3√3) / 2)
x = √((-5 - 3√3) / 2)
Learn more about U-substitution at:
brainly.com/question/21054456
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