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(21E+) Completing the Square $ax^2+bx+c\rightarrow a\left(x-h\right)^2+k$

Now we will convert $f\left(x\right)=2x^2+12x+14$ to the form $f\left(x\right)=a\left(x-h\right)^2+k$. Fill in the blanks.

$f\left(x\right)=2x^2+12x+14$

$f\left(x\right)=$ $\left(x^2+6x+7\right)$
(take out a factor of $\square$)

$f\left(x\right)=2\bigl(x^2+6x\phantom{AAA}+7\phantom{AAAI}\bigl)$
(move $+7$ to the right)

$f\left(x\right)=2\bigl(x^2+6x+$ $+7-$ $\bigl)$
(add and subtract)
$\phantom{f\left(x\right)=2\bigl(}\underbrace{\phantom{.............}}_{\phantom{AAAAAAA}}$
$f\left(x\right)=2\bigl\{\bigl(x+$ $\bigl)^2-$ $\bigl\}$
(complete the square)

$f\left(x\right)=2\bigl(x+3\bigl)^2-$
(distribute $2$ to inside the $\bigl\{ \bigl\}$ brackets)


Write $f\left(x\right)=3x^2-12x+2$ in the form $f\left(x\right)=a\left(x-h\right)^2+k$. Fill in the blanks.

$f\left(x\right)=3x^2-12x+2$

$f\left(x\right)=$ $\left(x^2-4x+\frac{2}{3}\right)$
(take out a factor of $\square$)

$f\left(x\right)=3\bigl(x^2-4x\phantom{AAA}+\frac{2}{3}\phantom{AAAI}\bigl)$
(move $+\frac{2}{3}$ to the right)

$f\left(x\right)=3\bigl(x^2-4x+$ $+\frac{2}{3}-$ $\bigl)$
(add and subtract)
$\phantom{f\left(x\right)=2\bigl(}\underbrace{\phantom{.............}}_{\phantom{AAAAAAA}}$
$f\left(x\right)=3\bigl\{\bigl(x-$ $\bigl)^2-$ $\bigl\}$
(complete the square) *use improper fractions

$f\left(x\right)=3\bigl(x-2\bigl)^2-$
(distribute $3$ to inside the $\bigl\{ \bigl\}$ brackets)


Write $f\left(x\right)=5x^2-15x-7$ in the form $f\left(x\right)=a\left(x-h\right)^2+k$. Fill in the blanks.

$f\left(x\right)=5x^2-15x-7$

$f\left(x\right)=$ $\bigl(x^2-3x-$ $\bigl)$
(take out a factor of $\square$) *use improper fractions

$f\left(x\right)=5\bigl(x^2-3x\phantom{AAA}-\frac{7}{5}\phantom{AAAI}\bigl)$
(move $-\frac{7}{5}$ to the right)

$f\left(x\right)=5\bigl(x^2-3x+$ $-\frac{7}{5}-$ $\bigl)$
(add and subtract)
$\phantom{f\left(x\right)=2\bigl(}\underbrace{\phantom{.............}}_{\phantom{AAAAAAA}}$
$f\left(x\right)=5\bigl\{\bigl(x-$ $\bigl)^2-$ $\bigl\}$
(complete the square) *use improper fractions

$f\left(x\right)=5\bigl(x-\frac{3}{2}\bigl)^2-$
(distribute $5$ to inside the $\bigl\{ \bigl\}$ brackets)


Write $f\left(x\right)=-2x^2-10x+5$ in the form $f\left(x\right)=a\left(x-h\right)^2+k$. Fill in the blanks.

$f\left(x\right)=-2x^2-10x+5$

$f\left(x\right)=$ $\bigl(x^2+5x-$ $\bigl)$
(take out a factor of $\square$) *use improper fractions

$f\left(x\right)=-2\bigl(x^2+5x\phantom{AAA}-\frac{5}{2}\phantom{AAAI}\bigl)$
(move $-\frac{5}{2}$ to the right)

$f\left(x\right)=-2\bigl(x^2+5x+$ $-\frac{5}{2}-$ $\bigl)$
(add and subtract)
$\phantom{f\left(x\right)=-2\bigl(}\underbrace{\phantom{.............}}_{\phantom{AAAAAAA}}$
$f\left(x\right)=-2\bigl\{\bigl(x+$ $\bigl)^2-$ $\bigl\}$
(complete the square) *use improper fractions

$f\left(x\right)=-2\bigl(x+\frac{5}{2}\bigl)^2+$
(distribute $-2$ to inside the $\bigl\{ \bigl\}$ brackets)


Practice

Write the following in the form $a\left(x-h\right)^2+k$. Give your answers using improper fractions.

$2x^2+16x-4$
$2x^2-16x+1$

$2x^2+14x-1$
$2x^2-18x-1$

$4x^2+16x+1$
$5x^2-20x-4$

$5x^2+25x+2$
$3x^2-9x+3$

$-x^2-6x-1$
$-x^2+10x-5$

$-x^2-3x+2$
$-x^2+x-3$

$-5x^2-20x+1$
$-4x^2+4x+5$