16.2 Solving by Factoring

16.2 Solving by Factoring

• 1.
  SOLVING QUADRATIC EQUATIONS by FACTORING!!! Section16.2
• 2.
  16.2 – SolvingQuadratic Equations by Factoring A quadratic equation is written in the Standard Form, 2 0ax bx c+ + = where a, b, and c are real numbers and .0a ≠ Examples: 2 7 12 0x x− + = 2 3 4 15x x+ = ( )7 0x x + = (standard form)
• 3.
  Zero Factor Property: Ifa and b are real numbers and if ,0ab = Examples: ( )7 0x x + = then or .0a = 0b = 0x = 7 0x + = 7x = −0x = 16.2 – Solving Quadratic Equations by Factoring
• 4.
  Zero Factor Property: Ifa and b are real numbers and if ,0ab = Examples: ( ) ( )10 3 6 0x x− − = then or .0a = 0b = 10 0x − = 3 6 0x − = 10x = 3 6x = 2x = 10 10 01 0x − =+ + 63 66 0x − + = + 3 6 3 3 x = 16.2 – Solving Quadratic Equations by Factoring
• 5.
  Solving Quadratic Equations: 1)Write the equation in standard form. 4) Solve each equation. 2) Factor the equation completely. 3) Set each factor equal to 0. 5) Check the solutions (in original equation). 16.2 – Solving Quadratic Equations by Factoring
• 6.
  REMEMBER FACTORING?!?!?! STEPS: 1. Factorout a GCF (Greatest Common Factor) 2. Choose your method of factoring: 2 terms – Difference of Squares  3 terms – a * c method  4 terms – Grouping 3x2 – 5x + 2 = 0 a*c = 3*2= 6 multiply to 6, 3x2 - 3x -2x +2=0 add to -5, -3 &-2 3x(x-1)-2(x-1)=0 (3x-2)(x-1) = 0 a b c
• 7.
  2 3 18 0xx− − = 6 0x − =3 0x + = ( )3x + 6x =3x = − 2 3 18x x− = 18:Factors of 1,18 2, 9 3, 6 ( ) ( ) 2 6 3 16 8− = 36 18 18− = 18 18= ( ) ( ) 2 13 3 83− −− = 9 9 18+ = 18 18= ( )6x − 0= 16.2 – Solving Quadratic Equations by Factoring
• 8.
  ( )3 18xx − = 18x = 2 3 18x x− = ( ) ( ) 2 18 13 18 8− = 324 54 18− = 270 18≠ ( ) ( ) 2 21 23 11 8− = 441 63 18− = 378 18≠ 3 18x − = 3 183 3x − =+ + 21x = If the Zero Factor Property is not used, then the solutions will be incorrect 16.2 – Solving Quadratic Equations by Factoring
• 9.
  2 4 5x x−= 1 0x + = 5 0x − = ( ) ( )1 5 0x x+ − = 1x = − 5x = ( )4 5x x − = 2 4 5 0x x− − = 16.2 – Solving Quadratic Equations by Factoring
• 10.
  2 3 7 6xx+ = 3 0x + = 3 2 0x − = ( ) ( )3 3 2 0x x+ − = 3x = − 2 3 x = ( )3 7 6x x + = 2 3 7 6 0x x+ − = 3 2x = 6:Factors of 2, 31, 6 3:Factors of 1, 3 16.2 – Solving Quadratic Equations by Factoring
• 11.
  2 9 24 16xx− = − 2 9 24 16 0x x− + = 3 4 0x − = ( ) ( )3 4 3 4 0x x− − = 4 3 x = 3 4x = ( )9 16and are perfect squares 16.2 – Solving Quadratic Equations by Factoring
• 12.
  3 2 18 0xx− = 2x 2 0x = 2x 3x = − 3 0x + = 3 0x − = 3x =0x = ( )2 9x − 0= ( )3x + ( )3x − 0= 16.2 – Solving Quadratic Equations by Factoring
• 13.
  ( ) ()2 3 3 20 7 0x x x+ − − = ( )3x + 3 0x + = 7x = 7 0x − = 3 1 0x + = 1 3 x = − 3x = − 3 1x = − 3:Factors of 1, 3 7 :Factors of 1, 7 ( )7x − 0=( )3 1x + 16.2 – Solving Quadratic Equations by Factoring
• 14.
  0 = A cliffdiver is 64 feet above the surface of the water. The formula for calculating the height (h) of the diver after t seconds is: 2 16 64.h t= − + How long does it take for the diver to hit the surface of the water? 0 = 0 = 2 0t + = 2 0t − = 2t = − 2t = seconds 2 16 64t− + 16− ( )2 4t − 16− ( )2t + ( )2t − 16.2 – Quadratic Equations and Problem Solving
• 15.
  2 x The square ofa number minus twice the number is 63. Find the number. ( )7x + 7x = − x is the number. 2 2 63 0x x− − = 7 0x + = 9 0x − = 9x = 2x− 63= 63:Factors of 1, 63 3, 21 7, 9 ( )9x − 0= 16.2 – Quadratic Equations and Problem Solving
• 16.
  ( )5 176ww+ = The length of a rectangular garden is 5 feet more than its width. The area of the garden is 176 square feet. What are the length and the width of the garden? ( )11w− The width is w. 11 0w− = 11w = The length is w+5.l w A× = 2 5 176w w+ = 2 5 176 0w w+ − = 16 0w+ = 16w = − 11w = 11 5l = + 16l = feet feet 176:Factors of 1,176 2, 88 4, 44 8, 22 11,16 ( )16w+ 0= 16.2 – Quadratic Equations and Problem Solving
• 17.
  x Find two consecutiveodd numbers whose product is 23 more than their sum? Consecutive odd numbers: x 5x = − 5x =2 2 2 25x x x+ = + 2 25 0x − = ( )5x + 5 0x + = 5 0x − = 5, 3− − 5, 7 5 2 3− + = − 5 2 7+ = 2.x + ( )2x + = ( )2x x+ + 23+ 2 22 2 2 25xx x x x+ = −+− 2 25 2525x =− − 2 25x = ( )5x − 0= 16.2 – Quadratic Equations and Problem Solving
• 18.
  a x= The lengthof one leg of a right triangle is 7 meters less than the length of the other leg. The length of the hypotenuse is 13 meters. What are the lengths of the legs? 12a = ( ).Pythagorean Th ( ) 22 2 7 13x x+ − = 5x = − 5= meters 7b x= − 13c = 2 2 14 49 169x x x+ − + = 2 2 14 120 0x x− − = ( )2 2 7 60 0x x− − = 2 5 0x + = 12 0x − = 12x = 12 7b = − meters 2 2 2 a b c+ = 60:Factors of 1, 60 2, 30 3, 20 4,15 5,12 ( )5x + ( )12x − 0= 6,10 16.2 – Quadratic Equations and Problem Solving