HP 33s 15–21, K 2 − a, J2 − a + y, Mathematics Programs 15–21

b0 = a0(4a2 – a32) – a12.

Let y0 be the largest real root of the above cubic. Then the fourth–order polynomial is reduced to two quadratic polynomials:

			x2 + (J + L)x + (K + M) = 0
			x2 + (J – L)x + (K – M) = 0
where J = a3/2
K = y0 /2
L =	J2 − a + y	0	⋅ (the sign of JK – a1/2)
	2
M =	K 2 − a
	0

Roots of the fourth degree polynomial are found by solving these two quadratic polynomials.

A quadratic equation x2 + a1x + a0 = 0 is solved by the formula

x1,2 = − a21 ± (a21 )2 − a0

If the discriminant d = (a1/2)2 – ao ≥ 0, the roots are real; if d < 0, the roots are complex, being u ± iv = −(a1 2) ± i − d .

Mathematics Programs 15–21