Question

Can someone please help me with this and show how they did it?

Answer 1

Now, a double root, or a root with a multiplicity of 2, means the root is there twice, namely, for a quadratic, that has a root of say "a", like x² + bx + c, the roots will be 0 = (x + a)(x + a), which produces a root of "a" twice, or a zero with a multiplicity of 2.

so. hmmmm let's use that, keeping in mind that, in the trinomial, the  middle term, as you know from factoring, is ax + ax, or 2ax.  The constant or last term is just the product of a * a, or a², thus


$\bf 0=(x+a)(x+a)\implies x^2+2ax+a^2 \\\\\\ \textit{but we know the coefficient of the middle term is }p-2 \\\\\\ therefore\qquad 2a~=~p-2\implies \boxed{2a+2=p}\\\\ -------------------------------\\\\ \textit{and we also know the constant is }3p+1 \\\\\\ therefore\qquad a^2=3p+1\implies a^2=3\left( \boxed{2a+2} \right)+1 \\\\\\ a^2=6a+6+1\implies a^2-6a-7=0 \\\\\\ (a-7)(a+1)=0\implies a= \begin{cases} 7\\ -1 \end{cases}$