Question

what is the derivation of the quadratic formula?

Answer 1

Question -

What is the derivation of the quadratic formula?

Solution -

Let the given quadratic equation be ax²+bx+c = 0 , a≠0

$\sf a {x}^{2} + bx + c = 0$

$\sf \hookrightarrow a {x}^{2} + bx = - c \: \: \: \: \: \: \: \: \: \: \: (transposing \: the \: constant \: term)$

$\sf \hookrightarrow {x}^{2} + \dfrac{b}{a} x = - \dfrac{c}{a} \: \: \: \: \: \: \: \: \: \: \: (dividing \: by \: the \: cofficient \: of \: {x}^{2} )$

Adding ${\sf {\dfrac{ {b}^{2} }{4 {a}^{2} } }}$ to both sides to make L.H.S. a perfect square

$\hookrightarrow\sf{x}^{2} + \dfrac{b}{a} x + \dfrac{ {b}^{2} }{4 {a}^{2} } = \dfrac{ {b}^{2} }{4 {a}^{2} } - \dfrac{c}{a}$

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

$\sf\hookrightarrow x + \dfrac{b}{2a} = \pm \dfrac{ \sqrt{ {b}^{2} - 4ac} }{2a} \: \: \: \: \: \: \: \: \: \: \: (taking \: the \: square \: root \: of \: both \: sides)$

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

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

${ \large{ \purple{ \boxed{ \dag \tt \:Hence \: proved \: \dag}}}}$