Exercise
Solve the equation:
5x^4+3x^2-8=0
Final Answer
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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