Question

g is a quadratic function where g(3) = 0, g( - 1) = 0, and g(0) = 0.36. Find an algebraic equation forg(u).

Answer 1

`\begin{gathered} \text{ Suppose we know that }g(u)=au^2+bu^{}+c,\text{ and that} \\ g(3)=0,g(-1)=0,\text{ and }g(0)=0.36,\text{ then this is equivalent to saying} \\ (3,0),(-1,0),(0,0.36)\text{ lie on the graph }g \\ \text{ These three points of the following equations} \\ 0=a(3)^2+b(3)+c\rightarrow0=9a+3b+c\text{ }(1) \\ 0=a(-1)^2+b(-1)+c\rightarrow0=a-b+c(2)_{}_{} \\ 0.36=a(0.36)^2+b(0.36)+c\rightarrow0.36=0.1296a+0.36b+c\text{ }(3) \\ \text{ Then subtract eq 1 from eq2 to get equation 4, then eq2 from eq3, and get eq5} \\ 8a+4b=0\text{ }(4) \\ 0.8704a-1.36b=-0.36\text{ }(5) \\ \text{ From equation 4 we get that} \\ 4b=-8a \\ b=-2a \\ \text{Substitute to eq 5} \\ 0.8704a-1.36(-2a)=-0.36 \\ 0.8704a+2.72a=-0.36 \\ 3.5904a=-0.36 \\ a=-(75)/(748) \\ \text{ Substitute it to eq1 get b and we get} \\ b=(75)/(374) \\ \text{ And since g(0)=0.36, then c=0.36 or} \\ c=(9)/(25) \\ g(u)=-(75)/(748)x^2+(75)/(374)x+(9)/(25) \end{gathered}`