# f(x) = x^4 - 4x^2 - 8 Use the graphing calculator to approximate the irrational solutions correct to 3 decimals.  If there is more than 1 real solution, enter them from smallest to largest.

embizze | High School Teacher | (Level 1) Educator Emeritus

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Find the real zeros for `f(x)=x^4-4x^2-8` :

The function is a quartic with positive leading coefficient -- as x tends to positive or negative infinity the function increases without bound. Since the function is a quartic, it can have at most 3 turning points.

The graph:

From the graph we see that there are 2 real zeros -- one between -3 and -2, the other between 2 and 3.

Using the calculate zero function we find the approximations are `x~~+-2.3375418`

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To 3 decimal places the zeros are `x~~2.338,x~~-2.338`

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For this function, we can calculate the exact values of the zeros. The function is quadratic in `x^2` so we can use the quadratic formula:

`x^2=(4+-sqrt(16-4(1)(-8)))/2`

`x^2=2+-2sqrt(3)`

`x=+-sqrt(2+-2sqrt(3))`

This gives all 4 roots -- `2-2sqrt(3)<0` so those roots are imaginary.