The Mystery Behind the Quadratic Formula

In Quadratic, there are lots of formula you should remember (or else it would be difficult for you) . I'm sure that you guys have already known this formula :

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

The question is, do you know that from the general form $ax^{ 2 }{ +bx }+c=0$ you can form the quadratic formula as shown above ? It seems not important for the beginners, but it really helps to improve our understanding in mathematics and we consider it as our general knowledge . Let me show you how it works :)

$Step\quad 1:$ Rearrange the general form in the term of $a{ x }^{ 2 }+{ bx }=-c$

$Step\quad 2:$ Since $a>0$, we can divide both side by $a$ to get ${ x }^{ 2 }+\frac { b }{ a } x=-\frac { c }{ a }$

$Step\quad 3:$ Since the coefficient of ${ x }^{ 2 }$ is 1, we can use the "Completing The Square" method to complete the square on the left side by adding the square of $\frac { 1 }{ 2 }$ of the coefficient of $x$ ; $(\frac { 1 }{ 2 } \times \frac { b }{ a } )^{ 2 }=\frac { { b }^{ 2 } }{ { 4a }^{ 2 } }$

$Step\quad 4:$ Add the product in $Step\quad 3$ to both side, then ${ x }^{ 2 }+\frac { b }{ a } x+\frac { { b }^{ 2 } }{ { 4a }^{ 2 } } =\frac { { b }^{ 2 } }{ { 4a }^{ 2 } } -\frac { c }{ a }$

$Step\quad 5:$ Factorise the left side completely to get $(x+\frac { b }{ 2a } )^{ 2 }$

$Step\quad 6:$ Solve the Right-side of the equation:

$=\frac { { b }^{ 2 } }{ { 4a }^{ 2 } } -\frac { c }{ a } \\ =\frac { { b }^{ 2 } }{ { 4a }^{ 2 } } -\frac { 4ac }{ { 4a }^{ 2 } } \\ =\frac { { b }^{ 2 }-4ac }{ { 4a }^{ 2 } }$

$Step\quad 7:$ Combine the Left-side and Right-side of the equation to get

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

You don't have to memorize all this 7 steps. All you need is to understand every solution in each step. Use your algebraic skills to solve the general form. Practice makes perfect. Trust me, it will improve your algebraic skills.