# Equations Formulas and Notes

Solving Equations

The general quadratic equation is ax^{2}-bx+c=0 . x represents an unknown, while a, b, and c are constants with a not equal to 0. Use the Quadratic formula to get the value of x.
Quadratic Formula: x = \frac{ - b \pm \sqrt{b^{2}-4ac} }{2a}

Example
Solve for x:    2x^{2}-3x-5=0
a = 2, b = -3, c = -5
x = \frac{ -b \pm \sqrt{b^{2}-4ac} }{2a}
x = \frac{ -(-3) \pm \sqrt{(-3)^{2}-4(2)(-5)} }{2(2)}
x = \frac{ 3 \pm \sqrt{(-3)^{2}-4(2)(-5)} }{2(2)}

x = \frac{ 3 \pm \sqrt{ 49 } }{ 4 }

x = \frac{ 3 \pm 7 }{ 4 }

x = \frac{ 3 + 7 }{ 4 } , x = \frac{ 3 - 7 }{ 4 }

x = 2.5 , x = -1

In \frac{ - b \pm \sqrt{ \Delta } }{2a}, \Delta = {b^{2}-4ac} .
If \Delta > 0 , there are two solutions.
If \Delta = 0 , there is only one solution.
If \Delta < 0 , there are no real solutions.

Factoring

\displaystyle x^2-y^2 = (x+y)(x-y)

\displaystyle x^3+y^3 = (x+y)(x^2-xy+y^2)

\displaystyle x^3-y^3 = (x-y)(x^2+xy+y^2

\displaystyle x^{2n}-y^{2n} = (x^n-y^n)(x^n+y^n)

\displaystyle (x+y)^2 = x^2+2xy+y^2

\displaystyle (x-y)^2 = x^2-2xy+y^2

\displaystyle (x+y)^3 = x^3+3x^2{y}+3x{y^2}+y^3

\displaystyle (x-y)^3 = x^3-3x^2{y}+3x{y^2}-y^3

\displaystyle (x+y)^4 = x^4+4x^{3}{y}+6x^{2}{y^2}+4xb^{3}+y^4

\displaystyle (x-y)^4 = x^4-4x^{3}{y}+6x^{2}{y^2}-4xb^{3}+y^4

\displaystyle (x+y)^5 = x^5+5x^{4}{y}+10x^{3}{y^2}+10x^{2}b^{3}+5xy^{4}+y^4 