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Document related concepts
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Transcript 
Simplify,
(a – x)(b – x)(c – x) . . . (z –x)
Simplify,
(a – x) . . . (x – x)(y – x)(z –x)
Simplify,
(a – x) . . . (x – x)(y – x)(z –x)
Simplify,
(a – x) . . . (x – x)(y – x)(z –x)
(a – x) . . . (0)(y – x)(z –x)
Simplify,
(a – x) . . . (x – x)(y – x)(z –x)
(a – x) . . . (0)(y – x)(z –x)
Any number times zero is zero.
Simplify,
(a – x) . . . (x – x)(y – x)(z –x)
(a – x) . . . (0)(y – x)(z –x)
Any number times zero is zero.
0
x2 + 9x + 20 = 0
x2 + 9x + 20 = 0
(x + 4)(x + 5) = 0
x2 + 9x + 20 = 0
(x + 4)(x + 5) = 0
(x + 4) = 0 (x + 5) = 0
x2 + 9x + 20 = 0
(x + 4)(x + 5) = 0
(x + 4) = 0 (x + 5) = 0
x = -4
x2 + 9x + 20 = 0
(x + 4)(x + 5) = 0
(x + 4) = 0 (x + 5) = 0
x = -4 or x = -5
The Zero Product Property:
If two or more numbers have a
product of zero, then at least one of
the numbers must be zero, or, for
any numbers a and b, if ab = 0, then
either a = 0 or b = 0.
3x2 + 15x = 0
3x2 + 15x = 0
3x(x + 5) = 0
3x2 + 15x = 0
3x(x + 5) = 0
3x = 0
3x2 + 15x = 0
3x(x + 5) = 0
3x = 0
(x + 5) = 0
3x2 + 15x = 0
3x(x + 5) = 0
3x = 0
x=0
(x + 5) = 0
3x2 + 15x = 0
3x(x + 5) = 0
3x = 0
x=0
(x + 5) = 0
x = -5
x – 16 = 0
2
x – 16 = 0
2
(x – 4)(x + 4) = 0
x – 16 = 0
2
(x – 4)(x + 4) = 0
(x – 4) = 0
x – 16 = 0
2
(x – 4)(x + 4) = 0
(x – 4) = 0
(x + 4) = 0
x – 16 = 0
2
(x – 4)(x + 4) = 0
(x – 4) = 0
x=4
(x + 4) = 0
x – 16 = 0
2
(x – 4)(x + 4) = 0
(x – 4) = 0 (x + 4) = 0
x = 4 or x = -4
x3 – 6x2 – 16x = 0
x3 – 6x2 – 16x = 0
x(x2 – 6x – 16) = 0
x3 – 6x2 – 16x = 0
x(x2 – 6x – 16) = 0
x(x – 8)(x + 2) = 0
x3 – 6x2 – 16x = 0
x(x2 – 6x – 16) = 0
x(x – 8)(x + 2) = 0
x=0
x3 – 6x2 – 16x = 0
x(x2 – 6x – 16) = 0
x(x – 8)(x + 2) = 0
x = 0 (x – 8) = 0
x3 – 6x2 – 16x = 0
x(x2 – 6x – 16) = 0
x(x – 8)(x + 2) = 0
x = 0 (x – 8) = 0 (x + 2) = 0
x3 – 6x2 – 16x = 0
x(x2 – 6x – 16) = 0
x(x – 8)(x + 2) = 0
x = 0 (x – 8) = 0 (x + 2) = 0
x=0
x3 – 6x2 – 16x = 0
x(x2 – 6x – 16) = 0
x(x – 8)(x + 2) = 0
x = 0 (x – 8) = 0 (x + 2) = 0
x = 0 or x = 8
x3 – 6x2 – 16x = 0
x(x2 – 6x – 16) = 0
x(x – 8)(x + 2) = 0
x = 0 (x – 8) = 0 (x + 2) = 0
x = 0 or x = 8 or x = -2
7x2 – 21x – 28 = 0
7x2 – 21x – 28 = 0
7(x2 – 3x – 4) = 0
7x2 – 21x – 28 = 0
7(x2 – 3x – 4) = 0
7(x + 1)(x – 4) = 0
7x2 – 21x – 28 = 0
7(x2 – 3x – 4) = 0
7(x + 1)(x – 4) = 0
(x + 1) = 0
7x2 – 21x – 28 = 0
7(x2 – 3x – 4) = 0
7(x + 1)(x – 4) = 0
(x + 1) = 0 (x – 4) = 0
7x2 – 21x – 28 = 0
7(x2 – 3x – 4) = 0
7(x + 1)(x – 4) = 0
(x + 1) = 0 (x – 4) = 0
x = -1
7x2 – 21x – 28 = 0
7(x2 – 3x – 4) = 0
7(x + 1)(x – 4) = 0
(x + 1) = 0 (x – 4) = 0
x = -1 or x = 4
x2 – 14x + 49 = 0
x2 – 14x + 49 = 0
(x – 7)(x – 7) = 0
x2 – 14x + 49 = 0
(x – 7)(x – 7) = 0
(x – 7) = 0
x2 – 14x + 49 = 0
(x – 7)(x – 7) = 0
(x – 7) = 0
(x – 7) = 0
x2 – 14x + 49 = 0
(x – 7)(x – 7) = 0
(x – 7) = 0
(x – 7) = 0
x= 7
16x2 – 36 = 0
16x2 – 36 = 0
4(4x2 – 9) = 0
16x2 – 36 = 0
4(4x2 – 9) = 0
4(2x – 3)(2x + 3) = 0
16x2 – 36 = 0
4(4x2 – 9) = 0
4(2x – 3)(2x + 3) = 0
2x – 3 = 0
16x2 – 36 = 0
4(4x2 – 9) = 0
4(2x – 3)(2x + 3) = 0
2x – 3 = 0
2x + 3 = 0
16x2 – 36 = 0
4(4x2 – 9) = 0
4(2x – 3)(2x + 3) = 0
2x – 3 = 0
2x = 3
2x + 3 = 0
16x2 – 36 = 0
4(4x2 – 9) = 0
4(2x – 3)(2x + 3) = 0
2x – 3 = 0
2x = 3
x = 1.5
2x + 3 = 0
16x2 – 36 = 0
4(4x2 – 9) = 0
4(2x – 3)(2x + 3) = 0
2x – 3 = 0
2x = 3
x = 1.5
2x + 3 = 0
2x = -3
16x2 – 36 = 0
4(4x2 – 9) = 0
4(2x – 3)(2x + 3) = 0
2x – 3 = 0
2x = 3
x = 1.5
or
2x + 3 = 0
2x = -3
x = -1.5
(18x3 – 50x)(x2 + 15x + 54) = 0
(18x3 – 50x)(x2 + 15x + 54) = 0
2x(9x2 – 25)(x + 9)(x + 6) = 0
(18x3 – 50x)(x2 + 15x + 54) = 0
2x(9x2 – 25)(x + 9)(x + 6) = 0
(18x3 – 50x)(x2 + 15x + 54) = 0
2x(9x2 – 25)(x + 9)(x + 6) = 0
2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0
(18x3 – 50x)(x2 + 15x + 54) = 0
2x(9x2 – 25)(x + 9)(x + 6) = 0
2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0
2x = 0
(18x3 – 50x)(x2 + 15x + 54) = 0
2x(9x2 – 25)(x + 9)(x + 6) = 0
2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0
2x = 0
3x – 5 = 0
(18x3 – 50x)(x2 + 15x + 54) = 0
2x(9x2 – 25)(x + 9)(x + 6) = 0
2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0
2x = 0
3x + 5 = 0
3x – 5 = 0
(18x3 – 50x)(x2 + 15x + 54) = 0
2x(9x2 – 25)(x + 9)(x + 6) = 0
2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0
2x = 0
3x + 5 = 0
x+9=0
3x – 5 = 0
(18x3 – 50x)(x2 + 15x + 54) = 0
2x(9x2 – 25)(x + 9)(x + 6) = 0
2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0
2x = 0
3x + 5 = 0
x+6=0
x+9=0
3x – 5 = 0
(18x3 – 50x)(x2 + 15x + 54) = 0
2x(9x2 – 25)(x + 9)(x + 6) = 0
2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0
2x = 0
x=0
3x + 5 = 0
x+6=0
x+9=0
3x – 5 = 0
(18x3 – 50x)(x2 + 15x + 54) = 0
2x(9x2 – 25)(x + 9)(x + 6) = 0
2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0
2x = 0
x=0
3x + 5 = 0
x+6=0
x+9=0
3x – 5 = 0
3x = 5
(18x3 – 50x)(x2 + 15x + 54) = 0
2x(9x2 – 25)(x + 9)(x + 6) = 0
2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0
2x = 0
3x + 5 = 0
x+6=0
x+9=0
3x – 5 = 0
x=0
3x = 5
3x = 5/3
(18x3 – 50x)(x2 + 15x + 54) = 0
2x(9x2 – 25)(x + 9)(x + 6) = 0
2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0
2x = 0
x=0
3x + 5 = 0
x+6=0
x+9=0
3x – 5 = 0
3x = -5
3x = 5
3x = 5/3
(18x3 – 50x)(x2 + 15x + 54) = 0
2x(9x2 – 25)(x + 9)(x + 6) = 0
2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0
2x = 0
x=0
3x + 5 = 0
x+6=0
x+9=0
3x – 5 = 0
3x = -5
3x = 5
3x = 5/3
3x = -5/3
(18x3 – 50x)(x2 + 15x + 54) = 0
2x(9x2 – 25)(x + 9)(x + 6) = 0
2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0
2x = 0
x=0
3x + 5 = 0
x+6=0
x+9=0
3x – 5 = 0
3x = -5
3x = 5
3x = 5/3
3x = -5/3
x = -9
(18x3 – 50x)(x2 + 15x + 54) = 0
2x(9x2 – 25)(x + 9)(x + 6) = 0
2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0
2x = 0
x=0
3x + 5 = 0
x+6=0
x+9=0
3x – 5 = 0
3x = -5
3x = 5
3x = 5/3
3x = -5/3
x = -9
x = -6
(18x3 – 50x)(x2 + 15x + 54) = 0
2x(9x2 – 25)(x + 9)(x + 6) = 0
2x(3x – 5)(3x + 5)(x + 9)(x + 6) = 0
2x = 0
x=0
3x + 5 = 0
x+6=0
x+9=0
3x – 5 = 0
3x = -5
3x = 5
3x = 5/3
x = -9
x = -6
3x = -5/3
x = -9, -6, -5/3, 0, 5/3
x2 – 15x = 50 – 10x
x2 – 15x = 50 – 10x
x2 – 5x – 50 = 0
x2 – 15x = 50 – 10x
x2 – 5x – 50 = 0
(x – 10)(x + 5) = 0
x2 – 15x = 50 – 10x
x2 – 5x – 50 = 0
(x – 10)(x + 5) = 0
(x – 10) = 0
x2 – 15x = 50 – 10x
x2 – 5x – 50 = 0
(x – 10)(x + 5) = 0
(x – 10) = 0 (x + 5) = 0
x2 – 15x = 50 – 10x
x2 – 5x – 50 = 0
(x – 10)(x + 5) = 0
(x – 10) = 0 (x + 5) = 0
x = 10
x2 – 15x = 50 – 10x
x2 – 5x – 50 = 0
(x – 10)(x + 5) = 0
(x – 10) = 0 (x + 5) = 0
x = 10 or x = -5
4x3 + x2 + 20 = 4x3 – 12x
4x3 + x2 + 20 = 4x3 – 12x
4x3 + x2 + 20 = 4x3 – 12x
x2 + 12x + 20 = 0
4x3 + x2 + 20 = 4x3 – 12x
x2 + 12x + 20 = 0
(x + 10)(x + 2) = 0
4x3 + x2 + 20 = 4x3 – 12x
x2 + 12x + 20 = 0
(x + 10)(x + 2) = 0
(x + 10) = 0
4x3 + x2 + 20 = 4x3 – 12x
x2 + 12x + 20 = 0
(x + 10)(x + 2) = 0
(x + 10) = 0 (x + 2) = 0
4x3 + x2 + 20 = 4x3 – 12x
x2 + 12x + 20 = 0
(x + 10)(x + 2) = 0
(x + 10) = 0 (x + 2) = 0
x = -10
4x3 + x2 + 20 = 4x3 – 12x
x2 + 12x + 20 = 0
(x + 10)(x + 2) = 0
(x + 10) = 0 (x + 2) = 0
x = -10 or x = -2
Some equations can be solved using
factoring. First perform some
transformations so that one side of
the equation is equal to zero. Then
factor the other side of the equation.
Use the Zero Product Property to
find all of the solutions to the
equation.
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