Solve Quartic with Casio FX115MS

11292018, 05:28 PM
(This post was last modified: 08282019 01:17 AM by Albert Chan.)
Post: #1




Solve Quartic with Casio FX115MS
y^4 + A y^2 + B y + C = 0
y^4 + 2D y^2 + D^2 = (2D  A) y^2  B y + (D^2  C) For some value of D, RHS can be made a perfect square: (y^2 + D)^2 = (2D  A) * (y  (B/2)/(2DA))^2 Let E = √(2DA), F = B/(2E), Quartic => (y^2 + E y + D+F)(y^2  E y + DF) = 0 Example: f(x) = x^4 + 2 x^3 + 3 x^2 + 4 x + 5 = 0 Depress the Quartic (remove cubic term) by substitute x = y + k, where k = b/(4a) We get depressed quartic: y^4 + A y^2 + B y + C k = b/(4a) = 2/4 = 0.5 C = f(k) = 3.5625 B = f(k−1) = 4.0625 A = f(k+1) = 8.0625 B = (A  B) / 2 = 2 A = A  1  B  C = 1.5 Find D such that RHS is perfect square, i.e. discriminant = B^2  4(2D  A)(D^2  C) = 0 We do not want complex values for E, so 2D ≥ A, D^2 ≥ C: D ≥ max(A/2, √C) ≈ max(0.75, 1.904) = 1.904 A trick to force Casio FX115MS to solve for D: 0ABC + B^2  4(2D  A)(D^2  C) D = 1.99130646 (Casio SOLVE, with guessed D = 2) E = √(2DA) = 1.575630959 F = B/(2E) = 0.6346663818 Quadratic Solver, enter a=1, b=+E, c=D+F, roots = −0.78781548 ± 0.857896758 i Quadratic Solver, enter a=1, b=−E, c=DF , roots = +0.78781548 ± 1.416093080 i x roots = y roots + k = 1.28781548 ± 0.857896758 i, 0.28781548 ± 1.416093080 i For calculator without Quad Solver, we can use this: y^2 + E y + (D+F) = 0 ==> y = −E/2 ± √(−F  D/2  A/4) y^2 − E y + (D−F) = 0 ==> y = +E/2 ± √(+F  D/2  A/4) Since C=(D+F)(DF), we might derive 1 from the other, hopefully better accuracy too. 

« Next Oldest  Next Newest »

Messages In This Thread 
Solve Quartic with Casio FX115MS  Albert Chan  11292018 05:28 PM
RE: Solve Quartic with Casio FX115MS  Albert Chan  11292018, 10:31 PM
RE: Solve Quartic with Casio FX115MS  Albert Chan  02042019, 01:39 PM
RE: Solve Quartic with Casio FX115MS  Albert Chan  08242019, 12:47 PM
RE: Solve Quartic with Casio FX115MS  SlideRule  03142020, 01:28 AM
RE: Solve Quartic with Casio FX115MS  Albert Chan  08302019, 02:02 PM
RE: Solve Quartic with Casio FX115MS  Albert Chan  09022019, 02:35 PM
RE: Solve Quartic with Casio FX115MS  Albert Chan  02012020, 12:27 PM
RE: Solve Quartic with Casio FX115MS  Albert Chan  03132020, 06:30 PM

User(s) browsing this thread: 1 Guest(s)