Linear-Quadratic Systems

1. $\text{Look}$ $\text{along}$ $\text{the}$ $\text{table}$ $\text{for}$ $\text{columns}$ $\text{where}$ $f(x\text{)}$ $\text{and}$ $g(x\text{)}$ $\text{are}$ $\text{equal.}$
2. $f(-4)=g(-4)=13$ $\text{and}$ $f(3)=g(3)=-8\text{.}$
3. $\text{So}$ $x=-4$ $\text{and}$ $x=3\text{.}$

Answer: $x=-4$ $\text{and}$ $x=3$

1. $\text{Substitute:}$ $x^2+25=-9x+5\text{,}$ $\text{so}$ $x^2+9x+20=0.$
2. $\text{Factor:}$ $(x+5)(x+4)=0\text{,}$ $\text{so}$ $x=-5$ $\text{and}$ $x=-4\text{.}$

Answer: $x=-5$ $\text{and}$ $x=-4$

1. $\text{Set}$ $\text{them}$ $\text{equal:}$ $-x^2+6x-9=0.$
2. $\text{That}$ $\text{is}$ $-(x-3{)}^2=0\text{,}$ $\text{so}$ $x=3$ $\text{only.}$
3. $\text{The}$ $\text{line}$ $\text{touches}$ $\text{the}$ $\text{parabola}$ $\text{once,}$ $\text{at}$ $(3,12)\text{.}$

Answer: $(3,12)$