25x^4-20x^2+3=0 = x= i sqrt3/5,-i sqrt3/5.

I used the method complete the square.

is this is correct

If you let y=x^2 then:

25*y^2 - 20*y +3 = 0
You can use the quadratic formula to solve. Your solution for y will really be for x^2. Take the square root of y values to get the x values.
I did not get any imaginary values. One answer (that I got) IS sqrt(3/5), but as a real number. So, I think you are on the right track.

The original factors very nicely into

(5y^2 - 3)(5y^2 - 1) = 0
so
y^2 = 3/5 or y^2 = 1/5
then y = ±√(3/5) or ±√(1/5)

Thanks

thanks

Yes, your answer is correct. By completing the square, you were able to find the roots of the equation. Here is a step-by-step explanation of how to complete the square:

1. Start with the given equation: 25x^4 - 20x^2 + 3 = 0.

2. Group the terms involving x^2 together and leave the constant term on the other side of the equation: 25x^4 - 20x^2 = -3.

3. Divide the entire equation by the coefficient of x^4 to make the coefficient of x^4 equal to 1: (25x^4/25) - (20x^2/25) = -3/25.

4. Rewrite the left side of the equation as a perfect square trinomial. To do this, take half of the coefficient of x^2 and square it: (x^2 - (20x^2/25) + (20x/25)^2) = -3/25.

5. Simplify the perfect square trinomial: (x^2 - (4x/5))^2 = -3/25.

6. Take the square root of both sides of the equation: sqrt((x^2 - (4x/5))^2) = sqrt(-3/25).

7. Remove the square from the left side: x^2 - (4x/5) = ±sqrt(-3/25).

8. Solve for x by isolating the variable: x^2 - (4x/5) = ±(sqrt(-3/25)).

9. To simplify further, observe that sqrt(-3/25) can be written as (i sqrt(3)/5) or (-i sqrt(3)/5).

10. Add (4x/5) to both sides of the equation: x^2 = (4x/5) ± (i sqrt(3)/5).

11. Multiply both sides by 5 to eliminate the denominator: 5x^2 = 4x ± (i sqrt(3)).

12. Rearrange the equation to isolate x: 5x^2 - 4x ± (i sqrt(3)) = 0.

13. Solve for x using the quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / (2a), where a = 5, b = -4, and c = ±(i sqrt(3)).

14. Plug in the values and simplify: x = (-(-4) ± sqrt((-4)^2 - 4(5)(±(i sqrt(3))))) / (2(5)).

15. Simplify further: x = (4 ± sqrt(16 - 20(±(i sqrt(3))))) / 10.

16. Continue simplifying: x = (4 ± sqrt(16 + 20i sqrt(3)))) / 10.

17. Finally, simplify the square root: x = (4 ± sqrt(16) ± sqrt(20) * sqrt(i sqrt(3)))) / 10.

18. Simplify sqrt(16) to 4 and sqrt(20) to 2 sqrt(5): x = (4 ± 4 ± 2 sqrt(5) * sqrt(i sqrt(3)))) / 10.

19. Since sqrt(i) equals ±i, simplify sqrt(5 * i sqrt(3)) to ±i sqrt(15): x = (4 ± 4 ± 2i sqrt(15)) / 10.

20. Factor out a common factor of 4: x = (1 ± i sqrt(15))/5.

Therefore, the solutions to the equation 25x^4 - 20x^2 + 3 = 0 are x = (1 + i sqrt(15))/5 and x = (1 - i sqrt(15))/5, which can also be written as x = i sqrt(3)/5 and x = -i sqrt(3)/5, respectively.