HOW TO COMPLETE THE SQUARE

Transcription

1 HOW TO COMPLETE THE SQUARE LANCE D. DRAGER. Introduction To complete the square means to take a quadratic function f() = a 2 + b + c and do the algebra to write it in the form f() = a( h) 2 + k, for some numbers h and k. For eample, f() = = 2( 2) 2 + (check it b working out the square on the right and simplifing). The advantage of doing this is that it allows ou to easil analze the graph of the quadratic function and its properties. Here is a brief summar of some of the properties ou can read off. Consider the function f() = a( h) 2 + k The graph is a parabola. In fact the graph = f() is the graph = a 2 shifted right h units and up k units. if a > the parabola opens up. If a < the parabola opens down. The verte of the parabola is the peak of the mountain if the parabola opens down, or the bottom of the pit if the parabola opens up. The verte is at the point (h, k). If the parabola opens down, the maimum value of the function f() is k, and this maimum occurs when = h. The range of the function is (, k]. There is no minimum value. If the parabola opens up, the minimum value of the function f() is is k, and this minimum occurs when = h. The range of the function is [k, ). There is no maimum value. The ais of smmetr of the parabola is the vertical line = h. To see these properties in action, consider Figure and Figure The Procedure We ll now go over a step b step procedure for completing the square, illustrated b the eample f() = Step : Write the function in decreasing powers of. In our eample, we write f() =

2 2 LANCE D. DRAGER 4 3 ais of smmetr 2 verte Figure. Graph of = 2( ) 2 + Step 2: Factor the coefficient of 2 out of the first two terms. In our eample, we write () f() = 3( 2 4) 8. Step 3: Find h and h 2. Inside the parentheses, ou ll have two terms that look like 2 + r. Set h = r/2 and calculate h 2. Looking at our eample in () we have 2 + r = 2 4, so r = 4. Thus, h = ( 4)/2 = 2 and h 2 = 2 2 = 4. Step 4: Add and subtract h 2 inside the parentheses. In our eample, we look at () and write f() = 3( ) 8 = 3( ) 8. new terms Step 5: Factor the first three terms in the parentheses. If we ve done it right, the first three terms in the parentheses are equal to ( h) 2. In our eample, h = 2, so we have f() = 3( ) 8 = 3(( 2) 2 4) 8. factor Step 6: Multipl the leading coefficient through the parentheses and simplif. In our eample, f() = 3(( 2) 2 4) 8 = 3( 2) 2 3( 4) 8 = 3( 2) = 3( 2) 2 +4.

3 HOW TO COMPLETE THE SQUARE 3 So, our final answer is f() = 3( 2) Done! 4 3 ais of smmetr verte Figure 2. Graph of = 2( + 2) Another eample, step b step. In this eample, we wind up with some fractions. Consider the function f() = Step : Write the function in decreasing powers of. This is alread done. Step 2: Factor the coefficient of 2 out of the first two terms. In the eample, we get f() = 2( 2 + 3) +. Step 3: Find h and h 2. In our eample, the coefficient of inside the parentheses is r = 3, so we get h = r/2 = 3/2 and so h 2 = 9/4. Step 4: Add and subtract h 2 inside the parentheses This gives us f() = 2( /4 9/4) +.

4 4 LANCE D. DRAGER Step 5: Factor the first three terms in the parentheses. We have ( h) = ( ( 3/2)) = ( + 3/2), and so f() = 2( /4 9/4) + = 2(( + 3/2) 2 9/4) +. (+3/2) 2 Step 6: Multipl the leading coefficient through the parentheses and simplif. In our problem, this gives so and we are done. f() = 2(( + 3/2) 2 9/4) + = 2( + 3/2) 2 9/2 + = 2( + 3/2) 2 9/2 + 2/2 = 2( + 3/2) 2 7/2, ( f() = ) An Application Problem Consider the cost and revenue functions from Problem 43 on page 73 of the tet. The functions are R() = (5.25) C() = 6 +, both functions have domain 4. These are the cost and revenue functions for a clock manufacturer. The variable represents the production level, in thousands of units, C() is the cost of manufacturing thousand units in thousands of dollars, and R() is the revenue (income from sales) that can be epected at a production level of thousand units, in thousands of dollars. Problem: find the production level that maimizes profit. Find the maimum profit. To find the solution, we write down the profit function P () = R() C(). Thus, we have P () = (5.25) (6+) = = To find the maimum of this function, we complete the square, following the steps above. Step : Write the function in decreasing powers of. We get P () = Step 2: Factor the coefficient of 2 out of the first two terms. This gives us P () =.25( 2 (4/.25)) 6 =.25( 2 32) 6 Step 3: Find h and h 2. In our problem, the coefficient of inside the parentheses is is r = 32, so h = r/2 = 6 and h 2 = 6 2 = 256. Step 4: Add and subtract h 2 inside the parentheses This gives us P () =.25( ) 6.

5 HOW TO COMPLETE THE SQUARE 5 Step 5: Factor the first three terms in the parentheses. We have ( h) = ( 6), and so P () =.25( ( 6) 2 256) 6 =.25(( 6) 2 256) 6. Step 6: Multipl the leading coefficient through the parentheses and simplif. We have P () =.25(( 6) 2 256) 6 =.25( 6) 2.25( 256) 6 =.25( 6) =.25( 6) We ve now completed the square to arrive at P () =.25( 6) To complete the problem, we note that the graph of P () is a parabola opening downward with verte (6, 6). Thus, the maimum profit occurs at a production level of = 6 thousand units. The profit at this production level will be 6 thousand dollars. As an aide in visualizing what the computations mean, Figure 3 shows the graph of the functions R(), C() and P (). 6 =R() =C() 4 2 verte (6, 6) 2 =P() Figure 3. The graphs for the application problem. Department of Mathematics and Statistics, Teas Tech Universit, Lubbock, TX address: drager@math.ttu.edu