new way to make quadratic equations easy

[Albert Chan]

(07-31-2021 10:19 AM)C.Ret Wrote:  ... The reduced quadratic equation is now : \( z^2+(-18+5i).z+(45-15i)=0 \)

We can also solve quadratics with half-angle formula, see (HP-67) Barkers's Equation

Let c = cot(θ/2)       → cot(θ) = (c²-1) / (2c)       → c² - 2*cot(θ)*c - 1 = 0

x² - 2*cot(θ)*x - 1 = (x - c)*(x + 1/c)

Let x = z/n, to scale constant term to -n², instead of -1.
In other words, solve for z² - 2*m*z - n² = 0

XCas> m,n := (-18.+5i)/-2, sqrt(-(45.-15i))
XCas> proot([1, -2m, -n*n])                                    → [3.0, 15.0-5.0*i]
XCas> -n*tan(atan(n/m)/2), n/tan(atan(n/m)/2)       → [3.0, 15.0-5.0*i]

Or, with "built-in" quadratic solver, asinh:

sinh(x) = (e^x - e^-x)/2       → (e^x)² - 2*sinh(x)*e^x - 1 = 0

XCas> -n/exp(asinh(m/n)), n*exp(asinh(m/n))         → [3.0, 15.0-5.0*i]

Since asinh(x) is odd function, z = ±n*exp(asinh(m/±n))

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We can also solve for z² - 2*m*z + n² = 0

XCas> m,n := (-18+5i)/-2., sqrt(45.-15i)
XCas> proot([1, -2m, n*n])                                    → [3.0, 15.0-5.0*i]
XCas> n*tan(asin(n/m)/2), n/tan(asin(n/m)/2)        → [3.0, 15.0-5.0*i]
XCas> n/exp(acosh(m/n)), n*exp(acosh(m/n))        → [3.0, 15.0-5.0*i]

Again, combine both roots, z = n*exp(±acosh(m/n))