Equations – Solving a polynomial equation – Exercise 5584

Exercise

Solve the equation

$x^4-3x^2+2=0$

Final Answer

Show 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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