Rewriting a Quadratic Expression
Task
 What is the minimum value taken by the expression $(x4)^2 + 6$? How does the structure of the expression help to see why?
 Rewrite the quadratic expression $x^2  6x  3$ in the form $(x\underline{\hspace{.5cm}})^2+\underline{\hspace{.5cm}}$ and find its minimum value.
 Rewrite the quadratic expression $2x^2 + 4x + 3$ in the form $\underline{\hspace{.5cm}}(x\underline{\hspace{.5cm}})^2+\underline{\hspace{.5cm}}$. What is its maximum value? Explain how you know.
IM Commentary
The goal of this task is to complete the square in a quadratic expression in order to find its minimum or maximum value. When we have a quadratic function $f(x)$ this corresponds to finding the vertex of its graph. Completing the square in a quadratic expression helps to find when the expression is equal to 0. For example, working with the expression from part (b), we have $$ x^2  6x  3 = (x3)^2  12. $$ We then set this equivalent expression equal to 0, $$ (x3)^2  12 = 0, $$ and solve by adding 12 to both sides of the equation and then extracting a square root. When applied to a general quadratic expression $ax^2 + bx + c$, completing the square leads to the quadratic formula.
Solution

The quantity $(x4)^2$ is never negative since it is the square of a real number. We have $(x4)^2 = 0$ only when $x  4 = 0$. So the expression $(x4)^2$ has a minimum value of 0 when $x = 4$. This means that the expression $(x4)^2 + 6$ has a minimum value when $x = 4$ and that minimum value is 6.
The structure of the expression $(x4)^2 + 6$ was vital in determining its minimum value as it allows us to focus on the simpler expression $(x4)^2$ and use the fact that the only real number whose square equals 0 is 0.

In order to write $x^2  6x  3$ as a perfect square plus a number, we focus on the first two terms $x^2  6x$. To write $x^2  6x$ as a perfect square plus a number note that, for any number $a$, $$(x+a)^2 = x^2 + 2ax +a^2.$$ We have $6x$ in the expression $x^2  6x$ so this means we want $2a = 6$ or $a = 3$. With this choice of $a$ we find $$ x^2  6x  3 = (x3)^2 12. $$
As in part (a), the minimum value will occur when $(x3)^2 = 0$ or $x = 3$. When $x = 3$ we see that the minimum value is 12.

For $2x^2 + 4x + 3$ we can work as in parts (a) and (b) but it is convenient to first factor out the leading coefficient of 2: $$ 2x^2 + 4x + 3 = 2\left(x^2  2x  \frac{3}{2}\right). $$ Alternatively, we could write $2x^2 + 4x +3 = 2(x^2 2x) + 3$. For the expression $x^2 2x  \frac{3}{2}$ we focus on $x^2  2x$ and, working as in part (b), we will want to choose $a = 1$ giving $$ x^2  2x  \frac{3}{2} = (x1)^2  \frac{5}{2}. $$ Substituting this into the above equality gives $$ 2x^2 + 4x + 3 = 2\left((x1)^2  \frac{5}{2}\right). $$ With the alternate expression, we would find $2x^2 +4x +3 = 2(x1)^2+5$, the expanded form of the right hand side of this equation (and the form requested in the question). The expression $2(x1)^2+5$ takes a maximum value when $(x1)^2 = 0$ or $x = 1$. When $x = 1$, this maximum value is $5$. We know that the value is a maximum because if $x \neq 1$ then $(x1)^2 \gt 0$ and so $2(x1)^2 \lt 0$ and the value of the expression is less than $5$.
Rewriting a Quadratic Expression
 What is the minimum value taken by the expression $(x4)^2 + 6$? How does the structure of the expression help to see why?
 Rewrite the quadratic expression $x^2  6x  3$ in the form $(x\underline{\hspace{.5cm}})^2+\underline{\hspace{.5cm}}$ and find its minimum value.
 Rewrite the quadratic expression $2x^2 + 4x + 3$ in the form $\underline{\hspace{.5cm}}(x\underline{\hspace{.5cm}})^2+\underline{\hspace{.5cm}}$. What is its maximum value? Explain how you know.