Algebraic Manipulation

Add/Subtract Algebraic Equations

$\frac{x+5}{x^2+5x+6}-\frac{x}{x^2-x-12}=0$ , Factorise the denominators

$\frac{x+5}{(x+2)(x+3)}-\frac{x}{(x-4)(x+3)}=0$ , Merge the two expressions

$\frac{(x+5)(x-4)-(x)(x+2)}{(x+2)(x+3)(x-4)}=0$ , Expand the numerator

$\frac{(x^2+x-20-x^2-2x)}{(x+2)(x+3)(x-4)}=0$ , Calculate

$\frac{(-x-20)}{(x+2)(x+3)(x-4)}=0$ , Tidy up

$\frac{(x+20)}{(x+2)(x+3)(4-x)}=0$

Rational Fractions

$\frac{3y}{4x-20}\div\frac{2x+10}{x^2-25}$ , Change multiplication sign into division

$\frac{3y}{4x-20}\div\frac{2x+10}{x^2-25}$ , Expand numerator & factorise denominator

$\frac{(3y)(x-5)(x+5)}{(8)(x-5)(x+5)}$ , Cancel out same factors

$\frac{3y}{8}$

Making x the Subject

$\frac{(3x^2+5)(3)}{4}+1=61$ , subtract 1 from both sides
$\frac{(3x^2+5)(3)}{4}=60$ , multiply 4 on both sides

$(3x^2+5)(3)=220$ , divide 3 out of both sides

$3x^2+5=80$ , subtract 5 on both sides

$3x^2=75$ , divide 3 out of both sides

$x^2=25$ , square root both sides

$x=5$

Proof for Quadratic Formula

$ax^2+bx+c=0$

$\frac{ax^2}{a}+\frac{bx}{a}=-\frac{c}{a}$

$(x+\frac{b}{2a})^2-(\frac{b}{2a})^2=-\frac{c}{a}$

$(x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}$

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

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