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Factorising a quadratic expression of the form $ax^2+bx+c$ by completing the square.

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Write the following quadratic expression in the form $a(x+b)^2+c$ by using the method of completing the square:

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\\[ \\simplify[unitFactor]{{a}x^2+{b}x+{c}} \\]

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For a quadratic expression of the form \\[ ax^2+bx+c,\\] when written in completed square form it becomes \\[ a\\left(x+\\frac{b}{2a}\\right)^2 - \\left(\\frac{b^2}{4a}\\right)+c.\\]

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So, \\[ \\begin{split} \\simplify[unitFactor]{{a}x^2+{b}x+{c}} &\\,= \\simplify[unitFactor,fractionNumbers]{{a}(x^2+{b/a} x) +{c}} \\\\ &\\,=\\simplify[unitFactor,fractionNumbers]{{a}((x+{b/(2a)})^2 - {(b/(2a))}^2) + {c}}, \\\\ &\\,=\\simplify[unitFactor,fractionNumbers]{{a}(x+{b/(2a)})^2 - {(b^2/(4a))} + {c}} \\\\&\\,=\\simplify[all,fractionNumbers]{{a}(x+{b/(2a)})^2 +{c-(b^2/(4a))}}. \\end{split} \\]

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