Completing the Square

Completing the Square (For help, go to Lessons 9-4 and 9-7.) Find each square. 1. (d – 4)22. (x + 11)2 3. (k – 8)2 Factor. 4.b2 + 10b + 25 5.t2 + 14t + 49 6. n2 – 18n + 81 Check Skills You’ll Need 10-5

Completing the Square 1. (d – 4)2 = d2 – 2d(4) + 42 = d2 – 8d + 16 2. (x + 11)2 = x2 + 2x(11) + 112 = x2 + 22x + 121 3. (k – 8)2 = k2 – 2k(8) + 82 = k2 – 16k + 64 4.b2 + 10b + 25 = b2 + 2b(5) + 52 = (b + 5)2 5.t2 + 14t + 49 = t2 + 2t(7) + 72 = (t + 7)2 6.n2 – 18n + 81 = n2 – 2n(9) + 92 = (n – 9)2 Solutions 10-5

2 The term to add to x2 – 16x is or 64. –16 2 Completing the Square Find the value of c to complete the square for x2 – 16x + c. The value of b in the expression x2 – 16x + c is –16. Quick Check 10-5

Completing the Square • First, write the left side of the equation as a perfect square. • X2 – 4x = 12 • X2 – 4x + 4 = 12 + 4 • Second, solve the equation by taking the square root of each side. • (x – 2)2 = V16 • X – 2 = ±4 • X = 4 + 2 and x = -4 + 2 • X = 6 and -2

Completing the Square • Did you see that this can be factored using two binomials?. • X2 – 4x = 12 • X2 – 4x – 12 = 0 • (x – 6)( x+ 2) = 0 • X = 6 and -2

Step 1: Write the left side of x2 + 5x = 50 as a perfect square. x2 + 5x = 50 2 2 Add , or , to each side of the equation. 25 4 5 2 5 2 x2 + 5x + = 50 + 200 4 25 4 = + 2 Rewrite 50 as a fraction with denominator 4. 5 2 2 2 2 5 2 5 2 5 2 x + x + Write x2 + 5x + as a square. 225 4 = Completing the Square Solve the equation x2 + 5x = 50. 10-5

Find the square root of each side. ± = 225 4 Simplify. 5 2 15 2 x + ± = 5 2 15 2 5 2 15 2 x + = Write as two equations. or x + = – 5 2 x + x = 5 or x = –10 Solve for x. Completing the Square (continued) Step 2: Solve the equation. Quick Check 10-5

Step 1: Rewrite the equation in the form x2 + bx = c and complete the square. x2 + 10x– 16 = 0 x2 + 10x = 16 Add 16 to each side of the equation. x2 + 10x+ 25 = 16 + 25 Add , or 25, to each side of the equation. 2 10 2 (x + 5)2 = 41 Write x2 + 10x +25 as a square. Completing the Square Solve x2 + 10x – 16 = 0 by completing the square. Round to the nearest hundredth. 10-5

x + 5 = ± 41 Find the square root of each side. x + 5 ± 6.40 Write as two equations. Use a calculator to find 41 x + 5 6.40 or x + 5 –6.40 x 6.40 – 5 or x –6.40 – 5 Subtract 5 from each side. Simplify x 1.40 or x –11.40 Completing the Square (continued) Step 2: Solve the equation. Quick Check 10-5

Define: width = x + 10 + 10 = x + 20 length = x + x + 10 + 10 = 2x + 20 Relate: length  width = area Write: (2x + 20)(x + 20) = 6000 2x2 + 60x + 400 = 6000 Step 1: Rewrite the equation in the form x2 + bx = c. 2x2 + 60x + 400 = 6000 2x2 + 60x = 5600 Subtract 400 from each side. x2 + 30x = 2800 Divide each side by 2. Completing the Square ALGEBRA 1 LESSON 10-5 Suppose you wish to section off a soccer field as shown in the diagram to run a variety of practice drills. If the area of the field is 6000 yd2, what is the value of x? 10-5

Step 2: Complete the square. x2 + 30x + 255 = 2800 + 225 (x + 15)2 = 3025 Write x2 + 30x + 255 as a square. Step 3: Solve each equation. Add , or 225, to each side. Take the square root of each side. (x + 15) = ± 3025 2 30 2 x + 15 = ± 55 Use a calculator. x + 15 = 55 or x + 15 = –55 x = 40 or x = –70 Use the positive answer for this problem. Completing the Square (continued) Quick Check The value of x is 40 yd. 10-5

Completing the Square Solve each equation by completing the square. If necessary, round to the nearest hundredth. 1.x2 + 14x = –43 2. 3x2 + 6x – 24 = 0 3. 4x2 + 16x + 8 = 40 –9.45, –4.55 –4, 2 –5.46, 1.46 10-5

Completing the Square