 # Equations – Solving a polynomial equation – Exercise 5588

Exercise

Solve the equation:

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

$$x=3, x=-1$$

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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