Equations – Solving a polynomial equation – Exercise 5588

Exercise

Solve the equation:

$x^6-26x^3=27$

Final Answer

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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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