10G (2.7) Solving by difference of square

10G (2.7)
Solving by difference of squares

Its easy to factor a quadratic when you have one square minus another square

square minus square

Multiple Choice

 

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor x² - 2²

• A = x , B = 2

  x² - 3²

  So

  ( x + 2)( x - 2)

• (x - 2)(x - 2)

Multiple Choice

 

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor x² - 4²

• ( x - 4)( x - 4)

• ( x - 2)( x - 2)

• ( x + 4)( x - 4)

• ( x + 4)( x + 4)

• ( x + 2)( x - 2)

Multiple Choice

Find the values of x for the previous question:

• x = -2, x = 2
• x = -5, x = 5
• x = -4, x = 4
• x = 0, x = 8
• x = -3, x = 3

Multiple Choice

 

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor x² - 5²

• (x + 25)
• (x - 25)
• ( x - 5)( x + 5)

• ( x + 5)( x + 5)

• ( x - 5)( x - 5)

Multiple Choice

Find the values of x for the previous question:

• x = 10, x = -10
• x = 2, x = -2
• x = 3, x = -3
• x = 7, x = -7
• x = 5, x = -5

Multiple Choice

 

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor x² - 7²

• (x - 14)
• (x - 49)
• (x + 49)
• ( x - 7)( x + 7)

• (x + 14)

Multiple Choice

Find the values of x for the previous question:

• x = 10, x = -10
• x = 5, x = -5
• x = 0, x = 14
• x = 1, x = 6
• x = 7, x = -7

Multiple Choice

 

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor x² - 9²

• ( x - 5)( x + 5)

• ( x - 12)( x + 12)

• ( x - 9)( x + 9)

• ( x - 6)( x + 6)

• ( x - 3)( x + 3)

Multiple Choice

Find the values of x for the previous question:

• x = 7, x = -7
• x = 9, x = -9
• x = 3, x = -3

• x = 5, x = -5
• x = 10, x = -10

Multiple Choice

 

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor x² - 4

• 4 = 2²

  so

  A = x , B = 2

  x² - 2²

  ( x + 2 )( x - 2 )

• ( x + 4 )( x - 4 )

Multiple Choice

Find the values of x for the previous question:

• x = 2, x = -2
• x = 1, x = 3
• x = -1, x = -3
• x = 5, x = -5
• x = 0, x = 4

Multiple Choice

 

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor x² - 16

• ( x + 2)( x - 8)

• ( x - 4)( x - 4)

• ( x + 16)( x - 1)

• ( x - 2)( x + 8)

• ( x + 4)( x - 4)

Multiple Choice

Find the values of x for the previous question:

• x = 0, x = 8
• x = 4, x = -4
• x = 6, x = -6
• x = 2, x = -2
• x = 3, x = -3

Multiple Choice

 

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor x² - 25

• ( x + 5)( x - 5)

• ( x + 10)( x - 2)

• ( x - 3)( x + 3)

• ( x - 7)( x + 3)

• ( x + 6)( x - 4)

Multiple Choice

Find the values of x for the previous question:

• x = 5, x = -5
• x = 2, x = -2
• x = 10, x = -10
• x = 7, x = -7
• x = 3, x = -3

Multiple Choice

 

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor x² - 64

• ( x + 10)( x - 6)

• ( x + 7)( x - 9)

• ( x + 8)( x - 8)

• ( x + 5)( x - 11)

• ( x + 12)( x - 4)

Multiple Choice

Find the values of x for the previous question:

• x = 0 or x = 16
• x = 10 or x = -10
• x = 8 or x = -8
• x = 4 or x = -4
• x = 12 or x = -12

Multiple Choice

 

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor x² - 100

• ( x + 100)( x - 100)

• ( x + 10)( x - 10)

• ( x + 50)( x - 50)

• ( x + 20)( x - 20)

• ( x + 12)( x - 12)

Multiple Choice

Find the values of x for the previous question:

• x = 15 or x = -15
• x = 5 or x = -5
• x = 0 or x = 20
• x = 10 or x = -10
• x = 8 or x = -8

Multiple Choice

 

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor 4² - x²

• A = 4 , B = x

  4² - x²

  So

  ( 4 + x)( 4 - x)

• ( x + 4 )( x - 4)

Multiple Choice

Find the values of x for the previous question:

• x = 3 or x = -3
• x = 4 or x = -4
• x = 2 or x = -2
• x = 0 or x = 8
• x = 5 or x = -5

Multiple Choice

 

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor 10² - x²

• ( 10 + x)( 10 - x)

• ( x - 10)( x + 10)

• ( 100 - x²)

• ( x + 10)( x + 10)

• ( 10 - x)( 10 - x)

Multiple Choice

Find the values of x for the previous question:

• x = 10 or x = -10
• x = 5 or x = -5

• x = 10

• x = 100
• x = 0

Multiple Choice

 

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor 13² - x²

• ( 13 - x)( 13 - x)

• ( 13 + x)( 13 - x)

• ( 13 - x)²

• ( x - 13)( 13 + x)

• ( 13 + x)²

Multiple Choice

Find the values of x for the previous question:

• x = 13

• x = 169
• x = 0
• $$x=\pm13$$

• x = 26 or x = -26

Multiple Choice

 

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor 49 - x²

• ( 7 - x)( 7 - x)

• ( 49 - x)( 49 + x)

• ( x - 7)( x + 7)

• ( 7 - x)( x - 7)

• ( 7 + x)( 7 - x)

Multiple Choice

Find the values of x for the previous question:

• x = 0
• $$x=\pm14\$$

• $$x=\pm7\$$

• x = 49
• x = -14

Multiple Choice

 

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor 81 - x²

• ( 9 - x)( 9 - x)

• ( x + 9)( x - 9)

• ( 81 - x²)

• ( 9 - x²)

• (9 - x)( 9 + x)

Multiple Choice

Find the values of x for the previous question:

• $$x=\pm\ 16$$

• $$x=\pm\ 4\$$

• $$x=\pm\ 10$$

• $$x=\pm\ 2$$

• $$x=\pm\ 8\$$

Multiple Choice

 

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor 121 - x²

• ( 13 - x)( 10 + x)

• ( x + 11)( x - 11)

• ( 12 - x)( 10 + x)

• ( 11 - x)( 11 + x)

• ( 11 - x)( 12 + x)

Multiple Choice

Find the values of x for the previous question:

• x = 0

• $$x=\pm\ 100$$

• $$x=\pm\ 1$$

• $$x=\pm\ 11$$

• $$x=\pm\ 10\$$

Multiple Choice

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor

$$4x^2-9$$

• A = 2x , B = 3

  ( 2x )² - 3²

  So

  ( 2x + 3)( 2x - 3)

• ( x + 3 )( x - 3 )

Multiple Choice

Find the values of x for the previous question:

• x = 0
• x = 1 or x = -1

• x = 2 or x = -2

• $$x=\pm\ 3\$$

• $$x\ =\ \pm\frac{3}{2}$$

Multiple Choice

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor

$$9x^2-16$$

• A = 3x , B = 4

  ( 3x )² - 4²

  So

  ( 3x + 4)( 3x - 4)

• ( x + 3 )( x - 3 )

Multiple Choice

Find the values of x for the previous question:

• $$x\ =\ \pm\frac{3}{4}$$

• x = -3, x = 3

• x = -5, x = 5
• $$x\ =\ \pm\frac{4}{3}$$

• $$x=\pm\ 4\$$

Multiple Choice

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor: $$36x^2-49$$

• ( 6x + 7)( 6x - 7)

• ( 3x + 7)( 12x - 7)

• ( 4x + 7)( 9x - 7)

• ( 6x + 8)( 6x - 8)

• ( 6x + 5)( 6x - 5)

Multiple Choice

Find the values of x for the previous question:

• x = 2
• $$x\ =\ \pm\frac{6}{7}$$

• x = 0

• $$x\ =\ \pm\frac{7}{6}$$

• $$x\ =\ \pm1$$

Multiple Choice

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor

$$16x^2-121$$

• ( 4x - 11)( 4x - 11)

• ( 16x - 121)

• ( 4x + 11)( 4x - 11)

• ( 4x + 11)( 4x + 11)

• ( 8x + 22)( 2x - 5)

Multiple Choice

Find the values of x for the previous question:

• x = 1, x = -1
• x = 16, x = -16
• $$x=\pm\frac{11}{4}$$

• $$x=\pm\ 4\$$

• $$x=\pm\frac{4}{11}$$

Multiple Choice

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor

$$64x^2-144$$

• 4( 8x + 3)( 8x - 3)

• 10( 5x + 3)( 5x - 3)

• ( 8x - 12)( 8x + 12)

  or better yet

  16( 2x - 3)( 2x + 3)

• ( 16x + 12)( 16x - 12)

  or better yet

  16( 4x + 3)( 4x - 3)

• 12( 6x + 3)( 6x - 3)

Multiple Choice

Find the values of x for the previous question:

• $$x=\pm\frac{2}{3}$$

• $$x=\pm\ 4\$$

• x = 2
• $$x=\pm\frac{3}{2}$$

• x = 0

Multiple Choice

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor

$$5x^2-20$$

• 20 = 5(4)

  so

  5( x² ) - 5(4)

  =

  5( x² ) - 5( 2² )

  =

  5( x² - 2² )

  so A = x , B = 2

  (5)( x + 2)( x - 2)

• ( x + 5 )( x - 5 )

Multiple Choice

Find the values of x for the previous question:

• x = -5, x = -1
• x = 4, x = -4

• x = -2, x = 2
• x = 1, x = 3
• x = 0, x = 5

Multiple Choice

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor

$$3x^2-75$$

• 3(x - 5)²

• 3( x + 5)( x + 5)

• 3( x - 5)( x - 5)

• 3( x + 5)( x - 5)

• 3( x - 5)( x + 5)

Multiple Choice

Find the values of x for the previous question:

• x = -5, 5
• x = -2, 2
• x = -7, 7
• x = -3, 3
• x = 0, 10

Multiple Choice

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor

$$2x^2-98$$

• 2( x + 3)( x - 3)

• 4( x + 7)( x - 7)

• 2( x + 10)( x - 10)

• 2( x + 7)( x - 7)

• 2( x + 14)( x - 14)

Multiple Choice

Find the values of x for the previous question:

• x = 0, 14
• x = -14, 0
• x = 1, 49
• x = -1, 49
• x = -7, 7

Multiple Choice

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$  

Factor

$$6x^2-96$$

• 6( x - 6)( x + 6)

• 6( x - 10)( x + 10)

• 6( x - 12)( x + 12)

• 6( x - 4)( x + 4)

• 6( x - 8)( x + 8)

Multiple Choice

Find the values of x for the previous question:

• x = 8, x = -8
• x = 12, x = -12
• x = 4, x = -4
• x = 10, x = -10
• x = 6, x = -6

Multiple Choice

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$   

Factor

$$9x^2-1089$$

• 9( x - 11)( x + 99)

• 9( x + 33)( x - 33)

• 9( x - 33)( x - 33)

• 9( x + 11)( x - 99)

• 9( x + 11 )( x - 11 )

Multiple Choice

Find the values of x for the previous question:

• x = -11, 11
• x = 0, 11
• x = -9, 9
• x = -10, 10
• x = -12, 12

Multiple Choice

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$   

Factor

$$24x^2-294$$

• 5( 4x + 5)( 4x - 5)

• 6( 2x + 7)( 2x - 7)

• 4( 6x + 3)( 6x - 3)

• 3( 8x + 21)( 8x - 21)

• 12x²-147

Multiple Choice

Find the values of x for the previous question:

• $$x\ =\ \pm\frac{2}{7}$$

• x = 0
• $$x=\pm\ 14\$$

• x = 3, x = -3

• $$x\ =\ \pm\frac{7}{2}$$

Multiple Choice

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$   

Factor

$$63x^2-343$$

• 7( 3x - 7)( 3x - 7)

• 7( 9x² - 49)

• 7( 9x² + 49)

• 7( 3x + 7)( 3x + 7)

• 7( 3x + 7)( 3x - 7)

Multiple Choice

Find the values of x for the previous question:

• $$x=\pm\ 7\$$

• $$x\ =\ \pm\frac{3}{7}$$

• x = 2
• $$x\ =\ \pm\frac{7}{3}$$

• $$x=\pm\ 10$$

Multiple Choice

$$A^2-B^{2\ }=\ \left(A+B\right)\left(A-B\right)$$   

Factor

$$405x^2-45$$

• 5( 9x + 3)(9x + 3)

• 5( 9x + 3)( 3x - 9)

• 5( 9x² - 3)

• 5( 27x² - 3)

• 5( 9x + 3 )( 9x - 3)

Multiple Choice

Find the values of x for the previous question:

• $$x=\pm\frac{3}{2}$$

• $$x=\pm3$$

• x = -5
• x = 0
• $$x=\pm\frac{1}{3}$$

Multiple Choice

Factor:

$$x^2+4$$

• (x + 2)(x - 2)

• It can't be factored

• (x + 4)(x - 4)

Multiple Choice

Can you find values for x:

$$x^2+4=0$$

• $$x\ =\ \pm\ 2$$

• No real solutions because

  $$x^2+4\ne\ne0$$

• $$x=\pm\ 4$$

It's pretty easy to factor a quadratic expression when it's the difference of perfect squares.

Next Time, we will try to do some stuff to make a quadratic equation into the difference of two perfect squares