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6.6 Quadratic Equations

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Multiplying Binomials A binomial has 2 terms Examples: x + 3, 3x – 5, x 2 + 2y 2, a – 10b To multiply binomials use the FOIL method (x + 3)(x + 4) = First Outer Inner LastFirst LastOuterInner

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Examples:Multiply (x + 5)(x + 6) (3x – 4)( 5x + 3)

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Common factors When factoring polynomials, first look for a common factor in each term Example: The binomial below has the factor 3 in each term 3x + 6y = (3)x + (3)2y = 3(x + 2y) To factor the above polynomial we used the distributive property. ac + bc = c(a + b)

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Examples: Factoring by distributive property 1.16n 2 + 12n 2.4x 2 +20x -12

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Examples: Difference of two squares a 2 –b 2 = (a + b)(a – b) 1.x 2 – 16 2.9y 2 - 25 3.49x 2 – 36 z 2 4.x 4 – 81 5.4x 2 y 2 – b 4 6.3x 3 – 12x

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Factoring Trinomials A trinomial has three terms. Example: x 2 + 5x + 6 If a trinomial factors, it factors into two binomials.

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Factoring trinomials First, look for a common factor. Then look for perfect square trinomials: a 2 + 2ab + b 2 = (a + b) 2 a 2 – 2ab + b 2 = (a – b) 2 If it is not a perfect square trinomial then factor into two binomials.

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Examples: Perfect square trinomials a 2 + 2ab + b 2 = (a + b) 2 a 2 – 2ab + b 2 = (a – b) 2 1.x 2 + 10x + 25 2.x 2 – 8y + 16 3.9x 2 – 24x + 16 4.25x 2 + 80xy + 64y 2 5.8x 2 y – 24xy + 18y

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Factoring trinomials of the form x 2 + bx + c x 2 + bx + c = (x + ___)(x + ____) x 2 - bx + c = (x - ___)(x - ____) To fill in the blanks look for factors of c that add up to equal b

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Examples: Factor. Check answers by FOIL x 2 -5x+6 x 2 +6x+8 x 2 -7x+10 x 2 +7x+12

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Factoring trinomials of the form x 2 + bx - c x 2 + bx - c = (x + ___)(x - ____) To fill in the blanks look for factors of c that subtract to equal b If the 1 st sign is negative place the larger factor with the negative sign If the 1 st sign is positive place the larger factor with the positive sign Signs will be different

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Examples: Factor x 2 +2x-35 x 2 -4x-12 x 2 -2x+15 x 2 +5x-36

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Factoring trinomials of the form ax 2 + bx + c One method is trial and error Try factors of a and c then FOIL to see if it works Examples: 2x 2 + 15x + 28 = ( + )( + ) 3x 2 +7x – 20= ( + )( - )

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Alternative method: Factor by grouping Factor by grouping is used to factor polynomials with 4 terms Example: Factor 10x 2 – 15x + 4x – 6 (10x 2 – 15x) + (4x – 6) 5x(2x – 3) + 2(2x – 3) (2x – 3)(5x + 2) Group together 1 st 2 terms and last 2 terms Factor out any common factors in each group Factor out (2x – 3) from each term

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Examples: Factor by grouping

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Factoring trinomials using factor by grouping. Since factor by grouping involves 4 terms we want to rewrite the trinomial as a polynomial with 4 terms

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General trinomials: ax 2 + bx + c Example: 2x 2 + 5x – 3 Multiply ac = 2(-3) = -6 Select 6 and -1 Factors of –6Sum of factors 1, -6-5 2, -3 3, - 21 6, -15*

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Example cont’d 2x 2 + (___ + ___) – 3 2x 2 + (6x + -1x) – 3 (2x 2 + 6x) + (-x – 3) 2x(x + 3) + -1(x + 3) (x + 3)(2x – 1)

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More examples 3x 2 – 14x – 5 15m 2 + 14m – 8 12x 2 + 23p + 5 12 – 20x – 13x 2

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Zero Product Property To solve a quadratic equation by factoring we will use the zero product property: If ab = 0, then a = 0 or b = 0 where a and b are any real numbers.

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Examples: Solve by factoring 1.x 2 – 3x – 28 = 0 2.x 2 + 4x = 12

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More examples 3.16x 2 = 49 4.4x 2 – 35x – 5 = 4

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More examples 5.x 2 + 6x = 13 = 4 6.4x 2 – 12x = 0

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Examples: Solve by factoring 1.x 2 – 3x – 28 = 0 2.x 2 + 4x = 12 3.4x 2 – 35x – 5 = 4 4.4x 2 – 12x = 0

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Examples: Solve by finding square roots 1.16x 2 = 49 2.5x 2 – 180 = 0 3.3x 2 = 24 4.x 2 – ¼ = 0

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Quadratic formula To use the quadratic formula, the equation must be in the form ax 2 + bx + c = 0

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Examples: solve using the quadratic formula 1.2y 2 + 4y = 30 2.x 2 – 7x + 1 = 0 3.5m 2 + 7m = -3 4.x 2 + 16 = 8x

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Discriminant The discriminant is b 2 – 4ac. The following chart describes the roots based on the value of the discriminant. b 2 – 4acRootsGraph > 02 realIntersects x-axis twice < 02 imaginaryDoes not intersect x-axis = 01 realIntersects x-axis once

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Examine the determinants in the previous examples to verify the chart

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