Rewrite the function by completing the square.

Question

$$f(x)=2x^2+13x+20$$
$$f(x)=$$
$$(x+$$
$${)}^2+$$

Answer 1

To rewrite the function $f(x)=2x^2+13x+20$ by completing the square, follow these steps:

1. Factor out the coefficient of $x^2$ (which is 2) from the first two terms: $$f(x)=2(x^2+\frac{13}{2}x)+20$$

2. Complete the square for the expression in parentheses $x^2+\frac{13}{2}x$. To do this, take half of the coefficient of $x$ (which is $\frac{13}{2}$), square it, and add and subtract it inside the parentheses.

   Half of $\frac{13}{2}$ is $\frac{13}{4}$. The square of $\frac{13}{4}$ is ${\left(\frac{13}{4}\right)}^2=\frac{169}{16}$.

   Then we can rewrite the expression: $$x^2+\frac{13}{2}x={(x+\frac{13}{4})}^2-\frac{169}{16}$$

3. Substitute this back into the function: $$f(x)=2({(x+\frac{13}{4})}^2-\frac{169}{16})+20$$

4. Distribute the 2: $$f(x)=2{(x+\frac{13}{4})}^2-2\cdot \frac{169}{16}+20$$ $$=2{(x+\frac{13}{4})}^2-\frac{338}{16}+20$$

5. Convert $20$ into sixteenths to combine the constants: $$20=\frac{320}{16}$$

6. Combine the constants: $$-\frac{338}{16}+\frac{320}{16}=-\frac{18}{16}=-\frac{9}{8}$$

7. Finally, we have the completed square form: $$f(x)=2{(x+\frac{13}{4})}^2-\frac{9}{8}$$

Thus, the completed square form is: $$f(x)=2{(x+\frac{13}{4})}^2-\frac{9}{8}$$