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[beautiful math coming... please be patient]
$ax^2+bx+c$
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[beautiful math coming... please be patient]
$=(ax^2+bx)+c$ |
(group first two terms) |
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[beautiful math coming... please be patient]
$=a(x^2+\frac{b}{a}x)+c$
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(factor $a\ne 0$ out of the first two terms) |
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[beautiful math coming... please be patient]
$=a\bigl(x^2+\frac{b}{a}x+{(\frac{b}{2a})}^2 -{(\frac{b}{2a})}^2\,\bigr)+c$ |
(add zero in an appropriate form inside the parentheses;
note that $\frac{b}{a}\div 2=\frac{b}{a}\cdot \frac{1}{2} = \frac{b}{2a}$)
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[beautiful math coming... please be patient]
$=a\bigl(x^2+\frac{b}{a}x+{(\frac{b}{2a})}^2\,\bigr)-a{(\frac{b}{2a})}^2 + c$ |
(distributive law) |
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[beautiful math coming... please be patient]
$=a{(x +\frac{b}{2a})}^2 + \text{stuff}$
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(rename as a perfect square) |
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