Exercise
Solve the equation:
x^6-26x^3=27
Final Answer
Solution
x^6-26x^3=27
In order to get a quadratic equation, we define a new variable:
y=x^3
We set the new variable:
y^2-26y-27=0
It’s a quadratic equation with the coefficients:
a=1, b=-26, c=-27
We solve it with the quadratic formula. Putting the coefficients in the formula gives us
y_{1,2}=\frac{26\pm \sqrt{{(26)}^2-4\cdot 1\cdot (-27)}}{2\cdot 1}=
=\frac{26\pm \sqrt{784}}{2}=
=\frac{26\pm 28}{2}
Hence, we get the solutions:
y_1=\frac{26+28}{2}=27
y_2=\frac{26- 28}{2}=-1
We go back to the original variable. From the first solution we get
27=x^3
x=3
From the second solution we get
-1=x^3
x=- 1
Finally, the solutions of the equation are
x=-1,3
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