We can rewrite quadratic equations given in the form $ax^2+bx+c$ as a square plus another term - this is called \"completing the square\".
\nThis can be useful when it isn't obvious how to fully factorise a quadratic equation.
\nRewrite the following expressions in the form \\[(x+b)^2-c\\] or \\[a(x+b)^2-c\\]
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\n\\[ (x+a)^2 = x^2 + 2ax + a^2 \\]
\nSo when we see an expression of the form $x^2 + 2ax$, we can rewrite it as $(x+a)^2-a^2$.
\n\nWe have $x^2+ \\var{evens1}x$, so we can replace it with $(x+\\var{evens1/2})^2-\\var{evens1/2}^2 = (x+\\var{evens1/2})^2 - \\var{evens1^2/4}$.
\nWe can check that this is equivalent to the original expression by expanding the brackets:
\n\\begin{align}
(x+\\var{evens1/2})^2 - \\var{evens1^2/4} &= \\simplify[basic]{ x^2 + 2*{evens1/2}*x + {evens1/2}^2 - {evens1^2/4} } \\\\
&= x^2 + \\var{evens1}x \\text{.}
\\end{align}
Replace $x^2 + \\var{odds}x$ by $\\simplify[basic]{(x+{odds}/2)^2-({odds}/2)^2}$, to obtain
\n\\begin{align}
x^2 + \\var{odds}x &= \\simplify[basic]{(x+{odds}/2)^2-({odds}/2)^2} \\\\[0.5em]
&= \\simplify[basic]{ (x+{odds}/2)^2 - {odds^2}/4} \\text{.}
\\end{align}
Replace $x^2+\\var{evens2}x$ with $(x+\\var{evens2/2})^2 - \\var{evens2/2}^2$.
\n\nNow remember that our original expression was not $x^2+\\var{evens2}x$ but $\\simplify[basic]{ x^2 + {evens2}x + {evens2-evens1}}$
\n\nHence:
\n\\begin{align}
\\simplify[basic]{ x^2 + {evens2}x + {evens2-evens1}} &= \\simplify[basic]{ (x+{evens2/2})^2 - {evens2/2}^2 + {evens2-evens1} } \\\\
&= \\simplify[basic]{ (x+{evens2/2})^2 + {evens2-evens1 - evens2^2/4} }
\\end{align}
First, notice that $\\simplify[basic]{ {all}x^2 + {multiall}x } = \\simplify[basic]{ {all}*( x^2 + {multiall/all} x)}$.
\nThen, we can replace $x^2 + \\var{multiall/all}x$ with $(x+\\var{multiall/all/2})^2 - \\var{multiall/all/2}^2$.
\n\\begin{align}
\\simplify[basic]{ {all}x^2 + {multiall}x + {odds3-evens2}} &= \\simplify[basic]{ {all}*( x^2 + {multiall/all} x) + {odds3-evens2}} & \\text{Extract the common factor of } \\var{all} \\\\
&= \\simplify[basic]{ {all}*( (x+{multiall/all/2})^2 - {multiall/all/2}^2) + {odds3-evens2} } & \\text{Complete the square}\\\\
&= \\simplify[basic]{ {all}*(x+{multiall/all/2})^2 - {all}*{(multiall/all/2)^2} + {odds3-evens2} } & \\text{Expand the constant term}\\\\
&= \\simplify[basic]{ {all}*(x+{multiall/all/2})^2 + {odds3-evens2 - (multiall/2)^2/all}} & \\text{Collect constants}
\\end{align}
It doesn't look like you've completed the square.
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It doesn't look like you've completed the square.
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It doesn't look like you've completed the square.
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