Answer:
Step-by-step explanation:
The quadratic expression in the standard form is given by :
f(x) = [tex]ax^{2}[/tex] + bx + c
(b) To complete the square:
Divide the equation through by a , the equation then becomes
f(x)= [tex]x^{2}[/tex] + [tex]\frac{bx}{a}[/tex] + [tex]\frac{c}{a}[/tex]
At this point, you are require to
(i) multiply the coefficient of x by 1/2
(ii) square the result
(iii) add the result to both sides , we have
f(x) = [tex]x^{2}[/tex] + [tex]\frac{bx}{a}[/tex] + [tex](\frac{b}{2a}) ^{2}[/tex] + [tex]\frac{c}{a}[/tex] + [tex](\frac{b}{2a}) ^{2}[/tex]by completing the square , we have
f(x) = [tex](x+\frac{b}{2a}) ^{2}[/tex] + [tex]\frac{c}{a}[/tex] - [tex]\frac{b^{2} }{4a^{2} }[/tex] . We did this in order to make the expression balance
(c) Using the order of operation to turn the expression back into standard form
i. Expand the function in the bracket , we have
f(x) = [tex]x^{2}[/tex] + [tex]\frac{bx}{a}[/tex] + [tex](\frac{b}{2a} )^{2}[/tex] + [tex]\frac{c}{a}[/tex] - [tex]\frac{b^{2} }{4a^{2} }[/tex]
⇒ f(x) = [tex]x^{2}[/tex] + [tex]\frac{bx}{a}[/tex] + [tex](\frac{b}{2a} )^{2}[/tex] + [tex]\frac{c}{a}[/tex] - [tex](\frac{b}{2a} )^{2}[/tex]
⇒f(x) = [tex]x^{2}[/tex] + [tex]\frac{bx}{a}[/tex] + [tex]\frac{c}{a}[/tex]
multiply through by the L.C.M , which is a , then we have
f(x) = [tex]ax^{2}[/tex] - bx + c .
Since the aim of completing the square is to make the expression a perfect square , then it will always result in a perfect square trinomial.
