9­5 Completing the Square perfect square (noun) ​
pur fikt skwehr Definition: ​
A perfect square is a number that can be written as the square of an integer. Example: ​
16 is a ​
perfect square​
because 16 = 42 . The expression x2 + 10x + 25 is a ​
perfect square 2
trinomial​
because x2 + 10x + 25 = (x + 5) . Nonexample: ​
27 is not a ​
perfect square​
because there is no integer that you can square to get 27. In previous lessons, you solved quadratic equations by finding square roots and by factoring. These methods work in some cases, but not all. Essential Understanding You can solve any quadratic by first writing it in the form m2 = n . you can model this process using algebra tiles. The algebra tiles below represent the expression x2 + 8x . Here is the same expression rearranged to form part of a square. Notice that the x­tiles have been split evenly into groups of four. You can complete the square by adding 42 , or 16 , 1­tiles. The completed square 2
is x2 + 8x + 16 , or (x + 4) . In general, you can change the expression x2 + bx into a perfect­square trinomial by adding ( 2b )2 to completing the x2 + bx . This process is called ​
square​
. The process is the same whether b is positive or negative. You can solve any quadratic equation by completing the square. This method turns every expression x2 + b x into a perfect­square trinomial. You ​
complete the square​
by adding ( 2b )2 to x2 + b x , where b ​
is the coefficient of the x­term. x2 + b x + ( 2b )2 = (x + 2b )2 Find the value of c such that each expression is a perfect­square trinomial. A.) x2 + 18x + c
B.) p2 − 30p + c
C.) g2 + 17g + c To solve an equation in the form x2 + bx + c = 0 , first subtract the constant term c from each side of the equation. Solve each equation by completing the square. If necessary, round to the nearest hundredth. D.) g2 + 7g = 144 E.) m2 + 16m =− 59 . F.) m2 + 12m + 19 = 0 2
The equation y = (x − h ) + k ​
represents a parabola with vertex ( h ​
, k ​
). You can use the method of completing the square to find the vertex of the quadratic functions of the form y = x2 + bx + c . Find the vertex of each parabola by completing the square. G.) y = x2 + 4x − 16 H.) y = x2 − 2x − 323 I.) y = x2 + 2x − 28 The method of completing the square works when a = 1 in ax2 + bx + c = 0 . To solve an equation when a =/ 1 , divide each side by a before completing the square. Solve each equation by completing the square. If necessary, round to the nearest hundredth. J.) 4a2 − 8a = 24 K.) 5n2 − 3n − 15 = 10 L.) 3r2 + 18r = 21 M.) Art The painting shown at the right has an area of 420 in. 2 . What is the value of x?