Minimizing and Maximizing a Quadratic
Example:
\[f(x)=x^2+6x-5\]
.Complete the square by writing in the form
\[(x+a)^2+b\]
.Expanding the brackets if this expression gives
\[ax^2+2ax+a^2+b\]
.Equate this to the quadratic:
\[x^2+6x-5=x^2+2ax+a^2+b\]
.Hence
\[2ax=6x \rightarrow a=3, \: -5=a^2+b \rightarrow b=-5-a^2=-5-3^2=-14\]
The completed square form is
\[(x+3)^2-15\]
.The minimum value of the quadratic is -14, occurring at
\[x+3=0 \rightarrow x=-3\]
. The function has no maximum (is unlimited).Example:
\[f(x)=10+8x-x^2\]
.Complete the square by writing in the form
\[a-(x+b)^2\]
.Expanding the brackets if this expression gives
\[a-x^2-2bx-b^2\]
.Equate this to the quadratic:
\[10+8x-x^2=a-x^2-2bx-b^2\]
.Hence
\[-2bx=8x \rightarrow b=-4, \: 10=a-b^2 \rightarrow a=10+b^2=10+(-4)^2=26\]
The completed square form is
\[26-(x-4)^2\]
.The maximum value of the quadratic is 26, occurring at
\[x-4=0 \rightarrow x=4\]
. The function has no minimum.Example:
\[f(x)=10+8x-2x^2\]
.Complete the square by writing in the form
\[a-b(x+c)^2\]
.Expanding the brackets if this expression gives
\[a-bx^2-2bcx-bc^2\]
.Equate this to the quadratic:
\[10+8x-2x^2=a-bx^2-2bccx-bc^2\]
.Hence
\[-2x^2=-bx \rightarrow b=2, \: 8x=-2bcx \rightarrow c=8/(-2b)=-2\]
\[10=a-bc^2 \rightarrow a=10+bc^2=10+2 \times (-2)^2=18\]
.The completed square form is
\[18-2(x-2)^2\]
.The maximum value of the quadratic is 18, occurring at
\[x-2=0 \rightarrow x=2\]
. The function has no minimum.
