Now, I will introduce quadratic formula.
Quadratic Formula
The answer of the quadratic equation, ( a x ^2 + bx + c = 0 ; ( a neq 0 ) ), is represented as below.
[ x = frac{ – b pm sqrt{ b^2 – 4ac}}{2a} ]Introduction
First, I directly introduce answers.
begin{eqnarray*} a x ^2 + bx + c & = & 0 x ^ 2 + frac{b}{a} x + frac{c}{a} & = & 0 left( x + frac{b}{2a} right) ^ 2 – frac{b^2}{4a^2} + frac{c}{a} & = & 0 left( x + frac{b}{2a} right) ^ 2 – frac{b^2 – 4ac}{4a^2} & = & 0 left( x + frac{b}{2a} right) ^ 2 & = & frac{b^2 – 4ac}{4a^2} end{eqnarray*}When right hand ( frac{b^2 – 4ac}{4a^2} ) is greater than than 0,
begin{eqnarray*} x + frac{b}{2a} & = & pm sqrt{frac{b^2 – 4ac}{4a^2}} & = & pm frac{sqrt{b ^ 2 – 4ac}}{2a} x & = & frac{ – b pm sqrt{b ^ 2 – 4ac}}{2a} end{eqnarray*}Detailed Introduction
Here, I write the same thing in more detail.
[ a x ^2 + bx + c = 0 ]Divide both sides by ( a ).
[ x ^2 + frac{b}{a} x + frac{c}{a} = 0 ]Denote ( frac{b}{a} = 2A ) , ( frac{c}{a} = B ) and rewrite the equation.
begin{eqnarray*} x^2 + 2A x + B & = & 0 x^2 + 2A x & = & -B end{eqnarray*}Then, add ( A^2 ) to both sides and the left side becomes square form. This is called “completing the squre”.
begin{eqnarray*} x^2 + 2A x + A^2 & = & A^2 – B left( x + A right) ^2 & = & A^2 – B end{eqnarray*}When ( A^2 – B geq 0 ),
begin{eqnarray*} x + A & = & pm sqrt{A^2 – B} x & = & – A pm sqrt{A^2 – B} end{eqnarray*}( 2A = frac{b}{a} ) , ( B = frac{c}{a} ) , hence replace ( A, B ) from the last equation and rewrite equation with ( a, b, c ).
begin{eqnarray*} x & = & – frac{b}{2a} pm sqrt{frac{b^2}{4a^2} – frac{c}{a}} & = & – frac{b}{2a} pm sqrt{frac{b^2}{4a^2} – frac{4ac}{4a^2}} & = & – frac{b}{2a} pm sqrt{frac{b^2 – 4ac}{4a^2}} & = & – frac{b}{2a} pm frac{sqrt{b^2 – 4ac}}{2a} & = & frac{ -b pm sqrt{b^2 – 4ac}}{2a} end{eqnarray*}
