Completing the Square/Examples

Examples of Completing the Square

Consider the quadratic equation:

$2 x^2 + 5 x + 1 = 0$

This is solved explicitly by completing the square as follows:

$\ds 2 x^2 + 5 x + 1$

$=$

$\ds 0$

$\ds \leadsto \ \ $

$\ds x^2 + \dfrac 5 2$

$=$

$\ds -\dfrac 1 2$

$\ds \leadsto \ \ $

$\ds \paren {x + \dfrac 5 4}^2$

$=$

$\ds -\dfrac 1 2 + \paren {\dfrac 5 4}^2$

$\ds $

$=$

$\ds -\dfrac 1 2 + \dfrac {25} {16}$

$\ds $

$=$

$\ds \dfrac {17} {16}$

$\ds \leadsto \ \ $

$\ds x + \dfrac 5 4$

$=$

$\ds \dfrac {\pm \sqrt {17} } 4$

$\ds $

$=$

$\ds \dfrac {-5 \pm \sqrt {17} } 4$