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Rewriting a Quadratic Expression


Alignments to Content Standards: A-SSE.B.3.b

Task

  1. What is the minimum value taken by the expression $(x-4)^2 + 6$? How does the structure of the expression help to see why?
  2. Rewrite the quadratic expression $x^2 - 6x - 3$ in the form $(x-\underline{\hspace{.5cm}})^2+\underline{\hspace{.5cm}}$ and find its minimum value.
  3. 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 $$ x^2 - 6x - 3 = (x-3)^2 - 12. $$ We then set this equivalent expression equal to 0, $$ (x-3)^2 - 12 = 0, $$ 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

  1. The quantity $(x-4)^2$ is never negative since it is the square of a real number. We have $(x-4)^2 = 0$ only when $x - 4 = 0$. So the expression $(x-4)^2$ has a minimum value of 0 when $x = 4$. This means that the expression $(x-4)^2 + 6$ has a minimum value when $x = 4$ and that minimum value is 6.

    The structure of the expression $(x-4)^2 + 6$ was vital in determining its minimum value as it allows us to focus on the simpler expression $(x-4)^2$ and use the fact that the only real number whose square equals 0 is 0.

  2. 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 $$ x^2 - 6x - 3 = (x-3)^2 -12. $$

    As in part (a), the minimum value will occur when $(x-3)^2 = 0$ or $x = 3$. When $x = 3$ we see that the minimum value is -12.

  3. 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 $$ x^2 - 2x - \frac{3}{2} = (x-1)^2 - \frac{5}{2}. $$ Substituting this into the above equality gives $$ -2x^2 + 4x + 3 = -2\left((x-1)^2 - \frac{5}{2}\right). $$ With the alternate expression, we would find $-2x^2 +4x +3 = -2(x-1)^2+5$, the expanded form of the right hand side of this equation (and the form requested in the question). The expression $-2(x-1)^2+5$ takes a maximum value when $(x-1)^2 = 0$ 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 $(x-1)^2 \gt 0$ and so $-2(x-1)^2 \lt 0$ and the value of the expression is less than $5$.

Shelbi says:

about 3 years

We have a similar task model in the Smarter Balanced assessment for this target, although I like the use of blanks in the IM version better than our current version which uses letters. We rejected the language of "complete the square" in the assessment probably for the same reason that this task avoids that exact language. Two years out of high school, students proficient in the mathematics involved in this task could still do the task as written, but probably would not know what we meant if we said "complete the square." The cluster level is an important organizer for the wording of such tasks -- as it really is about writing expressions in equivalent forms to solve problems.

Cam says:

about 3 years

Agreed! And thanks for your thoughts -- I would be very interested to see if there's any information/research/opinions as to the relative merits to using blanks over variables names. I can imagine pros and cons to each (intuitiveness vs. developing algebraic reasoniong skills and fluency).

Shelbi says:

about 3 years

Should (c) have another blank? ___ (x - )^2 + _

Michael Nakamaye says:

about 3 years

Indeed it should-- thank you!