# Algebra

## Sum and Product of Roots

The quadratic formula

$x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$

give the roots of a quadratic equation which may be real or imaginary. The ± sign in the radical indicates that

$x_1 = \dfrac{-b + \sqrt{b^2-4ac}}{2a}$ and $x_2 = \dfrac{-b - \sqrt{b^2-4ac}}{2a}$

where x_{1} and x_{2} are the roots of the quadratic equation ax^{2} + bx + c = 0. The sum of roots x_{1} + x_{2} and the product of roots x_{1}·x_{2} are common to problems involving quadratic equation.

## Derivation of Quadratic Formula

The roots of a quadratic equation *ax*^{2} + *bx* + *c* = 0 is given by the quadratic formula

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The derivation of this formula can be outlined as follows:

- Divide both sides of the equation
*ax*^{2}+*bx*+*c*= 0 by*a*. - Transpose the quantity
*c*/*a*to the right side of the equation. - Complete the square by adding
*b*^{2}/ 4*a*^{2}to both sides of the equation. - Factor the left side and combine the right side.
- Extract the square-root of both sides of the equation.
- Solve for
*x*by transporting the quantity*b*/ 2*a*to the right side of the equation. - Combine the right side of the equation to get the quadratic formula.

See the derivation below.

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