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Find a so that ax^2 + 15x + 4 + 3x + 5 is the square of a binomial.

 Jun 13, 2021
 #1
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ax^2 + 18x + 9   =        (3x+3)^2          when a = 9

 Jun 14, 2021
 #2
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The expression $ax^2+15x+4+3x+5$ simplifies to $ax^2+18x+9$.  Since it's the square of a binomial, it must equal $(cx+d)^2$ for some constants $c$ and $d$.  Now $(cx+d)^2= c^2x^2+2cdx+d^2$ and we can equate its coefficients to that of $ax^2+18x+9$ like so:
\begin{eqnarray*}
a &=& c^2 \\
2cd &=& 18 \\
d^2 &=& 9
\end{eqnarray*}
We solve this system of 3 equations in 3 unknowns $a$, $c$, and $d$.   Note that there are two solutions but both solutions have the same value of $a$.

 Jun 14, 2021

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