## Useful equations -Algebra Formulas

### 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

$x=–\frac{b}{a}$

ax2 + bx + c = 0

${x}_{1,2}=\frac{–b±\sqrt{{b}^{2}–4ac}}{2a}$

D = b2 – 4ac

### 4. Viete’s Formulas:

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

${x}_{1,2}=±\sqrt{–\frac{c}{a}}$

### 7. Cubic Equation. Cardano’s Formula:

y3 + py + q = 0

y1 = u + v

${y}_{2,3}=–\frac{1}{2}\left(u+v\right)±\frac{\sqrt{3}}{2}\left(u+v\right)i$

where,

$u=\sqrt[3]{–\frac{q}{2}+\sqrt{\left(\frac{q}{2}{\right)}^{2}}+\left(\frac{p}{3}{\right)}^{2}}$
$u=\sqrt[3]{–\frac{q}{2}–\sqrt{\left(\frac{q}{2}{\right)}^{2}}+\left(\frac{p}{3}{\right)}^{2}}$