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Equations – Solving a polynomial equation – Exercise 5581

Exercise

Solve the equation:

5x^4+3x^2-8=0

Final Answer


x=\pm 1

Solution

5x^4+3x^2-8=0

In order to get quadratic equation, we define a new variable:

y=x^2

We set the new variable:

5y^2+3y-8=0

It’s a quadratic equation with the coefficients:

a=5, b=3, c=-8

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 5\cdot (-8)}}{2\cdot 5}=

=\frac{-3\pm \sqrt{169}}{10}=

=\frac{-3\pm 13}{10}

Hence, we get the solutions:

y_1=\frac{-3+ 13}{10}=\frac{10}{10}=1

y_2=\frac{-3- 13}{10}=\frac{-16}{10}=-1.6

We go back to the original variable. From the first solution we get

1=x^2

x=\pm 1

From the second solution we get

-1.6=x^2

This equation has no real solution, because the left side is negative, while the right side is positive for every x.

Hence, the only solution of the equation is

x=\pm 1

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