Completing the Square
Completing the Square is the process of turning a quadratic relation in Standard Form (\(y = ax^2 + bx + c\)) into:
- \(x\) represents the independent variable
- \(y\) represents the dependent variable
- \(h\) represents the \(x\)-coordinate of the vertex
- \(k\) represents the \(y\)-coordinate of the vertex
The core advantage to completing the square is that you are able to determine the vertex of the parabola by identifying the \(h\) and \(k\) values. For example, we can determine that the vertex of the function \((x - 1)^2 + 5\) is \((1, 5)\) since \(h = 1\) and \(k = 5\). This function can be graphed as such:
We can identify \(d = 3\) and \(e = 4\). As a result, we can identify the vertex as \(\boldsymbol{(3, 4)}\)
Before graphing, we can create a table of values to identify the main points:
| x Values | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| y Values | 22 | 12 | 6 | 4 | 6 | 12 | 22 |
We can then sketch the corresponding graph as such:
Process for Completing the Square
- Factor the \(a\) value (if any) out of the first 2 terms
- Find the special value \(\left(\cfrac{b}{2}\right)^2\)
- Add, the special number to the bracket
- Subtract the expression by the special number by placing it outside the brackets and multiplying it by \(a\)
- Simplify the constants outside the brackets
- Factor the trinomial inside the brackets as the square of a binomial
Example
Rewrite the quadratic expression \(y = 2x^2 - 12x + 22\) by completing the square.
As \(a = 2\), we can factor this value out of the first 2 terms:
Next, as \(b = -6\), we can use this value to find the special number:
We can add the special number to the brackets and subtract the expression by the special number multiplied by factor \(a\). We can then simplify the constants:
\(y = 2(x^2 - 6x + 9) + 22 - (2)(9)\)
\(y = 2(x^2 - 6x + 9) + 22 - 18\)
\(y = 2(x^2 - 6x + 9) + 4\)
We can factor the trinomial inside the brackets by finding \(2\) numbers that result in the sum of the second term and the product of the third term:
\(y = 2(x^2 - 3x - 3x + 9) + 4\)
\(y = 2(x(x - 3) -3(x - 3)) + 4\)
\(y = 2(x - 3)^2 + 4\)
Therefore, we can determine that completing the square of function \(y = 2x^2 - 12x + 22\) results in \(\boldsymbol{y = 2(x - 3)^2 + 4}\).
As \(a = 1.5\), we can factor this value out of the first 2 terms:
Next, as \(b = 4\), we can use this value to find the special number:
We can add the special number to the brackets and subtract the expression by the special number multiplied by factor \(a\). We can then simplify the constants:
\(y = 1.5(x^2 + 4x + 4) - 5 - (1.5)(4)\)
\(y = 2(x^2 + 4x + 4) - 5 - 6\)
\(y = 2(x^2 + 4x + 4) - 11\)
We can factor the trinomial inside the brackets by finding \(2\) numbers that result in the sum of the second term and the product of the third term:
\(y = 1.5(x^2 + 2x + 2x + 4) - 11\)
\(y = 1.5(x(x + 2) + 2(x + 2)) - 11\)
\(y = 1.5(x + 2)^2 - 11\)
Therefore, we can determine that completing the square of function \(y = 1.5x^2 + 6x - 5\) results in \(\boldsymbol{1.5(x + 2)^2 - 11 = 0}\).
We can also determine that the function has a minimum at \(\boldsymbol{(-2, -11)}\) since \(a\) is positive.
Before graphing, we can create a table of values to identify the main points:
| x Values | -5 | -4 | -3 | -2 | -1 | 0 | 1 |
|---|---|---|---|---|---|---|---|
| y Values | 2.5 | -5 | -9.5 | -11 | -9.5 | -5 | 2.5 |
We can now sketch our graph:
Angie sold \(1200 \; [\text{tickets}]\) for the holiday concert at \($20/[\text{ticket}]\). Her committee is planning to increase the prices this year. Their research shows that for each \($2\) increase in ticket, \(60\) fewer tickets will be sold.
- Determine the revenue revelation that describes the ticket sales
- What should the selling price be to maximize revenue?
- How many tickets will be sold at the maximum revenue?
i. First, we can use the Revenue formula to describe the ticket price:
\(\text{Revenue} = \text{Price} \times \text{Quantity}\)
OR
\(R = P \times Q\)
Next, we can determine the respective expressions for Price and Quantitiy and combine them to determine the Revenue formula:
\(P = 20 + 2x\)
\(Q = 1200 - 60x\)
\(R = (20 + 2x)(1200 - 60x)\)
Therefore, we can determine that the revenue revelation describing the ticket sales is \(\boldsymbol{R = (20 + 2x)(1200 - 60x)}\).
ii. In order to find the maximum revenue (or max point) for the ticket sales, we must complete the square.
We can expand the expression using distributive property, collect like terms and rearrange in descending order:
\(R = (20)(1200) + (20)(-60x) + (2x)(1200) + (2x)(-60x)\)
\(R = 24000 - 1200x + 2400x - 120x^2\)
\(R = -120x^2 + 1200x + 24000\)
Next, as \(a = -120\), we can factor that value out of the first \(2\) terms:
Then, as \(b = -10\), we can use this value to determine the special number:
We can add the special number to the brackets and subtract the expression by the special number multiplied by factor \(a\). We can then simplify the constants:
\(R = -120(x^2 - 10x + 25) + 24000 - (-120)(25)\)
\(R = -120(x^2 - 10x + 25) + 24000 + 3000\)
\(R = -120(x^2 - 10x + 25) + 27000\)
We can factor the trinomial inside the brackets by finding \(2\) numbers that result in the sum of the second term and the product of the third term:
\(R = -120(x^2 - 5x - 5x + 25) + 27000\)
\(R = -120(x(x - 5) -5(x - 5)) + 27000\)
\(R = -120(x - 5)^2 + 27000\)
As \(x = 5\), we can plug this value into the Price formula to determine what the new ticket prices are:
\(P = 20 + 2x\)
\(P = 20 + (2)(5)\)
\(P = $30\)
Therefore, we can determine that the selling price to maximize revenue is \(\boldsymbol{$}\textbf{30/[ticket]}\).
iii. We can use \(x = 5\) to determine the maximum amount of tickets that will be sold using the Quantity formula:
\(Q = 1200 - 60x\)
\(Q = 1200 - (60)(5)\)
\(Q = 900\;[\text{tickets}]\)
Therefore, we can determine that we can sell \(\textbf{900 [tickets]}\) at an adjusted price of \($30\).
iv. As \(x = 5\), we can plug this value into the Revenue formula to determine what the new ticket prices are:
\(R = (20 + 2x)(1200 - 60x)\)
\(R = (20 + (2)(5))(1200 - (60)(5))\)
\(R = (20 + 10)(1200 - 300)\)
\(R = (30)(900)\)
\(R = $27000\)
Therefore, we can determine that the max revenue will be \(\boldsymbol{$27000}\).