(21E+) Completing the Square $x^2+bx+c\rightarrow \left(x-h\right)^2+k$
Fill in the blanks.
$x^2+10x+\square =\left(x+5\right)^2$
$x^2+18x+\square =\left(x+9\right)^2$
Fill in each blank. Separate your answers with a comma. For example:
$x^2+4x+\boxed{ \text{A} } =\left(x+\boxed{ \text{B} }\,\right)^2\quad$ Ans. $4,2$
$x^2+12x+\boxed{ \text{A} } =\left(x+\boxed{ \text{B} }\,\right)^2$
$x^2+6x+\boxed{ \text{A} } =\left(x+\boxed{ \text{B} }\,\right)^2$
$x^2+16x+\boxed{ \text{A} } =\left(x+\boxed{ \text{B} }\,\right)^2$
$x^2+18x+\boxed{ \text{A} } =\left(x+\boxed{ \text{B} }\,\right)^2$
Fill in the blanks.
$x^2-12x+\square =\left(x-6\right)^2$
$x^2-6x+\square =\left(x-3\right)^2$
Fill in each blank. Separate your answers with a comma. For example:
$x^2-6x+\boxed{ \text{A} } =\left(x- \boxed{ \text{B} }\,\right)^2\quad$ Ans. $9,3$
$x^2-10x+\boxed{ \text{A} } =\left(x-\boxed{ \text{B} }\,\right)^2$
$x^2-12x+\boxed{ \text{A} } =\left(x-\boxed{ \text{B} }\,\right)^2$
$x^2-16x+\boxed{ \text{A} } =\left(x-\boxed{ \text{B} }\,\right)^2$
$x^2-18x+\boxed{ \text{A} } =\left(x-\boxed{ \text{B} }\,\right)^2$
Fill in the blanks. Give your answer as an improper fraction.
$x^2-3x+\square =\left(x-\frac{3}{2}\right)^2$
$x^2+5x+\square =\left(x+\frac{5}{2}\right)^2$
Fill in each blank. Separate your answers with a comma. Give your answers as an improper fractions. For example:
$x^2-5x+\boxed{ \text{A} } =\left(x- \boxed{ \text{B} }\,\right)^2\quad$ Ans. $\frac{25}{4},\frac{5}{2}$
$x^2+9x+\boxed{ \text{A} } =\left(x+\boxed{ \text{B} }\,\right)^2$
$x^2-9x+\boxed{ \text{A} } =\left(x-\boxed{ \text{B} }\,\right)^2$
$x^2-3x+\boxed{ \text{A} } =\left(x-\boxed{ \text{B} }\,\right)^2$
$x^2-7x+\boxed{ \text{A} } =\left(x-\boxed{ \text{B} }\,\right)^2$
Now we will convert $f(x)=x^2+6x+7$ to the form $f(x)=\left(x-h\right)^2+k$.
$f(x)=x^2+6x+7$
$f(x)=x^2+6x\phantom{++|}+7$
$f(x)=x^2+6x+$
$+7-$
$\phantom{f(x)=\,}\underbrace{\phantom{x^2+6x+9}}_{\left(x+3\right)^2}$
$f(x)=\left(x+3\right)^2-$
$\phantom{}$
$\left(\text{move}+7\;\text{to the right}\right)$
$\left(\text{add and subtract}\right)$
$\phantom{}$
Write $f\left(x\right)=x^2-4x-3$ in the form $f\left(x\right)=\left(x-h\right)^2+k$.
$f(x)=x^2-4x-3$
$f(x)=x^2-4x\phantom{++|}-3$
$f(x)=x^2-4x+$
$-3-$
$\phantom{f(x)=\,}\underbrace{\phantom{x^2-4x-3}}$
$f(x)=\bigl(x-$
$\bigl)^2-$
$\phantom{}$
$\left(\text{move}-3\;\text{to the right}\right)$
$\left(\text{add and subtract}\right)$
$\phantom{}$
Write $f(x)=x^2+10x+11$ in the form $f(x)=\left(x-h\right)^2+k$.
$f(x)=x^2+10x+28$
$f(x)=x^2+10x\phantom{++|}+28$
$f(x)=x^2+10x+$
$+28-$
$\phantom{f(x)=\,}\underbrace{\phantom{x^2+10x+25}}$
$f(x)=\bigl(x+$
$\bigl)^2+$
$\phantom{}$
$\left(\text{move}+28\;\text{to the right}\right)$
$\left(\text{add and subtract}\right)$
$\phantom{}$
Write $f(x)=x^2+3x+5$ in the form $f(x)=\left(x-h\right)^2+k$. Give your answers as improper fractions.
$f(x)=x^2+3x+5$
$f(x)=x^2+3x\phantom{++|}+5$
$f(x)=x^2+3x+$
$+5-$
$\phantom{f(x)=\,}\underbrace{\phantom{x^2+3x+\frac{9}{4}}}$
$f(x)=\bigl(x+$
$\bigl)^2+$
$\phantom{}$
$\left(\text{move}+5\;\text{to the right}\right)$
$\left(\text{add and subtract}\right)$
$\phantom{}$
Practice
Write the following in the form $\left(x-h\right)^2+k$. Give your answers using improper fractions.
$x^2+6x+5$
$x^2+4x+2$
$x^2-4x-1$
$x^2-8x-4$
$x^2+5x+3$
$x^2+3x-2$
$x^2-5x-5$
$x^2-7x-4$