Useful equations

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

Solutions : x₁, x₂, y₁, y₂, y₃

1. Linear Equation in One Variable:

ax + b = 0

$$ x = - \frac ba $$

2. Quadratic Equation:

ax² + bx + c = 0

$$ x_{1,2} = {-b \pm \sqrt{b^2-4ac} \over 2a} $$

3. Discriminant:

D = b² – 4ac

4. Viete’s Formulas:

If x² + px + q = 0, then

x₁ + x₂ = -p and

x₁x₂ = q

5. Quadratic Equation:

ax² + bx = 0

$$ x_1=0, x_2 = \frac {-b}{a} $$

6. Quadratic Equation:

ax² + c = 0

$$ x_{1,2} = \pm \sqrt{-\frac ca} $$

7. Cubic Equation. Cardano’s Formula:

y³ + py + q = 0

y₁ = u + v

$$ y_{2,3} = - \frac 12 (u+v) \pm \frac {\sqrt{3}}{2} (u+v)i $$

where,

$$ u = \sqrt[3]{ - \frac q2 + \sqrt{( \frac q2 )^2} + ( \frac p3 )^2 } $$

$$ u = \sqrt[3]{ - \frac q2 - \sqrt{( \frac q2 )^2} + ( \frac p3 )^2 } $$