[tex]\qquad \textit{perfect square trinomial} \\\\ (a\pm b)^2\implies a^2\pm \stackrel{\stackrel{\text{\small 2}\cdot \sqrt{\textit{\small a}^2}\cdot \sqrt{\textit{\small b}^2}}{\downarrow }}{2ab} + b^2 \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ x^2 \pm \stackrel{\stackrel{2\sqrt{x^2}\sqrt{c^2}}{\downarrow }}{6x} +c^2~\hspace{10em}6x=2\sqrt{x^2}\sqrt{c^2}\implies 6x=2xc \\\\\\ \cfrac{6x}{2x}=c\implies 3=c\implies 9=c^2~\hfill \boxed{(x\pm 3)^2}[/tex]
c=9 makes the given expression a perfect square trinomial.
The given expression is:
[tex]x^{2} +6x+c[/tex]
We can rewrite it as:
[tex]x^{2} +2*x*3 + c[/tex].....(1)
The expansion of [tex](a+b)^2[/tex] is [tex]a^{2} +2ab+b^{2}[/tex]
Comparing (1) with [tex]a^{2} +2ab+b^{2}[/tex]
We get
a=x
b=3
c=b²
So, c=3² =9.
So, c=9 makes the given expression a perfect square trinomial.
Hence, c=9 makes the given expression a perfect square trinomial.
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