Complete the equation to show two equivalent expressions.g2 – 4g – 21 = (g – )(g + )

Accepted Solution

Answer:[tex]g^2-4g-21=(g-7)(g+3)[/tex]Step-by-step explanation:To complete the left side of the equation, we need to bring it to the form [tex](g-a)(g+b)[/tex]expanding this expression we get: [tex]g^2+bg-ag-ab[/tex][tex]g^2+(b-a)g-ab[/tex]Thus we have [tex]g^2-4g-21=g^2+(b-a)g-ab[/tex]from here we see that for both sides of the equation to be equal, it must be that[tex]b-a=-4[/tex][tex]-ab=-21[/tex].Getting rid of the negative signs we get:[tex]a-b=4[/tex][tex]ab=21[/tex]At this point we can either guess the solution to this system (that's how you usually solve these types of problems) or solve for [tex]a[/tex] and [tex]b[/tex] systematically.The solutions to this set are [tex]a=7[/tex] and [tex]b=3[/tex]. (you have to guess on this—it's easier)Therefore, we have [tex](g-a)(g+b)=(g-7)(g+3)[/tex]which completes our equation[tex]\boxed{ g^2-4g-21=(g-7)(g+3)}[/tex]