Completing the square (1)
- 1. M ODELING C OMPLETING THE S QUARE Use algebra tiles to complete a perfect square trinomial. 2 Model the expression x + 6x. x2 x x x x x x Arrange the x-tiles to form part of a square. x 1 1 1 To complete the square, x 1 1 1 add nine 1-tiles. x 1 1 1 You have completed the square. x2 + 6x + 9 = (x + 3)2
- 2. S OLVING BY C OMPLETING THE S QUARE To complete the square of the expression x2 + bx, add the square of half the coefficient of x. 2 2 b = x+ b x2 + bx + ( ) ( 2 2 )
- 3. Completing the Square What term should you add to x2 – 8x so that the result is a perfect square? S OLUTION 2 –8 The coefficient of x is –8, so you should add to the expression. ( 2) , or 16, 2 –8 2 x – 8x + 2 ( ) = x2 – 8x + 16 = (x – 4)2
- 4. Completing the Square Factor 2x2 – x – 2 = 0 S OLUTIO N 2x2 – x – 2 = 0 Write original equation. 2x2 – x = 2 Add 2 to each side. 21 x – 2 x=1 Divide each side by 2. 2 1 1 2 1 2 1 2 x – 1 2 x+ – 1 ( ) 4 =1+ 1 16 Add – ( • 2 2 ) ( 4) = – , or 16 to each side.
- 5. Completing the Square 2 1 1 2 1 2 1 2 x – 1 2 x+ – 1 4 ( ) =1+ 1 16 Add – (• 2 2 ) ( 4) = – , or 16 to each side. 2 1 17 (x – 4 ) = 16 Write left side as a fraction. x– 1 = ± 17 Find the square root of each side. 4 4 1 ± 17 Add 1 to each side. x= 4 4 4 1 17 1 17 The solutions are + ≈ 1.28 and – ≈ – 0.78. 4 4 4 4
- 6. C HOOSING A S OLUTION M ETHOD Investigating the Quadratic Formula Perform the following steps on the general quadratic equation ax2 + bx + c = 0 where a ≠ 0. ax2 + bx = – c Subtract c from each side. bx –c Divide each side by a. x2 + a + = a bx b 2 –c b 2 2 x + a + ( ) 2a ( ) = a + 2a Add the square of half the coefficient of x to each side. b x + ) = –a + b c2 2 ( 2a 4a 2 Write the left side as a perfect square. b 2 – 4ac + b 2 Use a common denominator to express ( x+ 2a ) = 2 4a the right side as a single fraction.
- 7. C HOOSING A S OLUTION M ETHOD Investigating the Quadratic Formula b 2 – 4ac + b 2 Use a common denominator to express (x+ 2a ) = 2 4a the right side as a single fraction. ± b − 4ac 2 b Find the square root of each side. x+ = 2a 2a Include ± on the right side. ± b2 − 4ac b Solve for x by subtracting the same x= – 2a 2a term from each side. –b ± b − 4ac 2 Use a common denominator to express x= the right side as a single fraction. 2a