Solving quadratic equations by completing a square

Solving quadratic equations by completing a square

• 1.
  Quadratics: Completing the square LO 11.2 4
• 2.
  Completing the square. Factorisethe following:- 1. x2 – 3x + 9 = ( x – 3)(x – 3) 2. x2 + 8x + 16 = ( x + 4)(x + 4) 3. 2x2 – 8x + 8 = 2(x2 – 4x + 4) = 2(x – 2)(x – 2) □ □ 4. x2 - 10x + 25 = ( x − 5 )( x − 5)
• 3.
  Completing the square Eg1 x2 - 8 x - 7 Space out = x2 – 8x -7 8 b /2 : add and = x2 - 8 x + ( 2)2 - 7 – ( 16 ) subtract same = x2 - 8 x + ( 4 )2 - 7 – 16 number =(x–4)2 - 23
• 4.
  Completing the square Eg1 NB coefficient of x 3x2 – 12x + 9 MUST be 1 = 3[ x2 – 4x + 3] Common factor = 3[x2 – 4x +3 ] Space out 4 b /2 : add and subtract = 3[x – 4 x + ( 2 ) + 3 – ( 4)] 2 2 same number. = 3[x – 4 x + ( 2 ) + 3 – 4] 2 2 = 3[ ( x – 2 ) 2 – 1] Multiply both terms =3(x–2)2 – 3 by 3
• 5.
  Completing the square. • Do the following:- 1. x2 + 6x + 1 2. x2 – 5x + 3 3. 2x2 + 8x – 4 4. 3x2 – 9x + 2
• 6.
  Completing the square •Solutions 1. x2 + 6x + 1 = x2 + 6x +(6/2)2 + 1 – 9 = (x + 3)2 - 8 2. x2 – 5x + 3 = x2 – 5x +( 5/2)2 + 3 – 25/4 = (x – 5/2 )2 + 3 – 61/4 = (x – 21/2)2 - 3 1/4
• 7.
  Completing the square. • Solutions cont. 3. 2x2 + 8x – 4 = 2[ x2 + 4x –2 ] = 2[ x2 + 4x +( 4/2)2 – 2 – 4] = 2[ ( x + 2)2 – 6] = 2( x + 2 )2 - 12
• 8.
  Completing the square Solutionscont 4. 3 x 2 – 9x – 2 = 3[ x2 – 3x – 2/ 3 ] = 3[ x2 – 3x + (3/2)2 – 2/3 – 9/4 ] = 3[ ( x – 3/2) 2 – 8/12 – 27/12 ] = 3[ ( x – 3/2 )2 – 35/12 ] = 3( x – 3/2 ) 2 – 35/4 .
• 9.
  Solving for x: x2 – 3x – 7 =0 x 2 – 3x =7 x2 - 3x + (3/2)2 = 7 + (9/4) ( x - 3/2 )2 = 28/4 + 9/4 ( x – 3/2 )2 = 37/4 37 x – 3/2 =± 4 3 37 3 ± 37 x = ± = 2 4 2
• 10.
  Solving for x 12x2 + 8x – 4 = 0 x2 + 4x – 2 =0 x2 + 4x =2 x2 + 4x + ( 4/2)2 =2+4 ( x + 2)2 = 6 x+2 =± 6 x = -2 ± 6
• 11.
  ax2 + bx+ c =0 c c c x2 + (b/a) x + − 0 = a − a a  b 2  b2  x + ( /a)x + 2 b   =  2-  4a  /a c  2a    b − 4ac 2 4a 2 ( x + b/2a ) 2 = ± b 2 − 4ac 2a x + b/2a b = b 2 − 4ac − b ± b − 4ac 2 − ± 2a 2a 2a x = =