Exercise
Solve the equation
x^4-3x^2+2=0
Final Answer
Solution
x^4-3x^2+2=0
In order to get a quadratic equation, we define a new variable:
y=x^2
We set the new variable:
y^2-3y+2=0
It’s a quadratic equation with the coeffients:
a=1, b=-3, c=2
We solve it with the quadratic formula. Putting the coefficients in the formula gives us
y_{1,2}=\frac{3\pm \sqrt{{(-3)}^2-4\cdot 1\cdot 2}}{2\cdot 1}=
=\frac{3\pm \sqrt{1}}{2}=
=\frac{3\pm 1}{2}
Hence, we get the solutions:
y_1=\frac{3+ 1}{2}=\frac{4}{2}=2
y_2=\frac{3- 1}{2}=\frac{2}{2}=1
We go back to the original variable. From the first solution we get
2=x^2
x=\pm \sqrt{2}
From the second solution we get
1=x^2
x=\pm 1
Finally, the solutions of the equation are
x=\pm 1,\pm\sqrt{2}
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