Equations Formulas

Equations Formulas

Real numbers: a, b, c, p, q, u, v

Solutions: x₁, x₂, y₁, y₂, y₃

1. Linear equation in One Variable

Ax + b = 0 , x = – $\frac{b}{a}$ .

2. Quadratic Equation

ax² + b x + c = 0 , x_1,2 = $\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$

3. Discriminate

D = b² -4ac

4. Viete’s Formulas

If x² + px +q = 0 , than

$\left\{\begin{matrix} x_{1}+x_{2}&=-p \\ x_{1}x_{2}& =q \end{matrix}\right.$

5. ax² +bx = 0 ,x₁ = 0 , x₂ = $-\frac{b}{a}$ .

6. ax²+ c = 0, x_1,2 = ± $\sqrt{\frac{-c}{a}}.$

7. Cubic Equation. Cardano’s Formula.

y³ + py +q = 0,

y_{1 =} u +v, y_{2,3 =} $\frac{-1}{2}(u + v) ± \frac{\sqrt{3}}{2}(u+v)i,$

where

$u = \sqrt[3]{\frac{-q}2{}+\sqrt{(\frac{q}{2})^{2}+(\frac{p}{3})^{2}}}$ , v = $\sqrt[3]{\frac{-q}{2}-\sqrt{(\frac{q}{2})^{2}+(\frac{p}{3})^{2}}}$ .