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Equations – Solving a polynomial equation – Exercise 5588

Exercise

Solve the equation:

x^6-26x^3=27

Final Answer


x=3, x=-1

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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