Useful equations -Algebra Formulas

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

Solutions : x₁, x₂, y₁, y₂, y₃

ax + b = 0

x = - \frac{b}{a}

ax² + bx + c = 0

x_{1, 2} = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

D = b² – 4ac

If x² + px + q = 0, then

x₁ + x₂ = -p and

x₁x₂ = q

ax² + bx = 0

x_{1} = 0, x_{2} = \frac{- b}{a}

ax² + c = 0

x_{1, 2} = \pm \sqrt{- \frac{c}{a}}

y³ + py + q = 0

y₁ = u + v

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

where,

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

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