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Equations – Solving a polynomial equation – Exercise 5584

Exercise

Solve the equation

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

Final Answer


x=\pm 1, x=\pm \sqrt{2}

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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