determine number of roots for: f ( x ) = x^4 − 26x^2 +10

Answer 1

y = x⁴ -26x²+10

Finding the roots mean we must make it all equal to 0

0 = x⁴ -26x²+10

We solve this problem by completing the square.

Half of 26 is 13. 13² = 169

We want our equation on our right to be:

x⁴ -26x²+169

so we need to add 159 to both sides.

159 = x⁴ -26x²+169

159 = (x²-13)²

from here we want to get x by itself.

±√159 = x² -13

±√159 + 13 = x²

x = ±√(13 ± √159)

This give you 4 different answers for x which is what we wanted as there are 4 roots.

Answer 2

same as the highest degree: 4, but some may repeat or be imaginary

x^4 -26x^2 + 10 = 0

let y = x^2

y^2 -26y + 10 = 0

calculate the discriminant

b^2 - 4ac = 26^2 -4(1)(10) > 0, if >0 there's 2 real solutions for y

but y = x^2, so there's 2x2 solutions = 4, for each y, x = + or - sqry