1.3 KiB
A numerically careful quadratic formula
A quadratic equation has the form ax^2 + bx + c = 0 with a \neq 0. The usual formula is
[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. ]
The quantity D = b^2 - 4ac is the discriminant. If D > 0 there are two real roots; if D = 0 there is a repeated root; if D < 0 there are no real roots.
When b is large compared with a and c, one of the two roots can lose digits to cancellation. A more stable pair of formulas is to first compute the root that does not subtract nearly equal quantities,
[ x_1 = \frac{-b - \mathrm{sign}(b)\sqrt{D}}{2a}, ]
and then recover the second root from the product of roots, x_2 = c / (a x_1).
Worked example
Solve 2x^2 - 7x - 15 = 0.
Here a = 2, b = -7, and c = -15. Then D = 49 + 120 = 169 = 13^2, so
[ x = \frac{7 \pm 13}{4}. ]
The roots are x = 5 and x = -3/2. Checking: 2(25) - 7(5) - 15 = 0, and 2(9/4) + 21/2 - 15 = 0. The same roots come from factoring: (2x + 3)(x - 5) = 0.
I keep a short set of algebra checks like this in my notes. When I want a second independent walkthrough I use the free solver at MathsSolverGPT. After any calculator or solver result, substitute the roots back into the original equation before you move on.