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Factoring ax2 + bx + c “Bottoms Up”

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6x2 + 13x + 6 Step 1: multiply the constant (c) term by the coefficient (a), of the leading term Constant is 6 Leading term is 6 Therefore 6 * 6 is 36

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6x2 + 13x + 6 Step 2: Rewrite the equation by replacing the constant with the number in step 1, and remove the leading coefficient. x2 + 13x + 36

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**Step 3: Factor the new equation from**

step 2: x2 + 13x +36 (x + 4)(x + 9)

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(x + 4)(x + 9) Step 4: Divide each constant term in the factored form by the original coefficient of the leading term. Original coefficient of the leading term: 6

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**Then reduce the fractions:**

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**Step 5: Now we use the “bottoms up” method- Move the denominator into the numerator to get:**

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Check it- use FOIL:

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Remember you may check some equations using graphing calculator- type in the equation in y = and find where it crosses the x-axis. Not all equations cross the x-axis. This particular one crosses at x = -2/3 and x = -3/2. Look at these solutions and the end result of step 4, and then look at step 5. Factoring leads to solutions, the zeros, the roots.

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10x2 - x - 3 Step 1: multiply the constant (c) term by the coefficient (a), of the leading term Constant is -3 Leading term is 10 Therefore -3 * 10 is -30

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10x2 - x - 3 Step 2: Rewrite the equation by replacing the constant with the number in step 1, and remove the leading coefficient. x2 - x - 30

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**(x + 5)(x - 6) x2 - x - 30 Step 3: Factor the new equation from**

step 2: x2 - x - 30 (x + 5)(x - 6)

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(x + 5)(x - 6) Step 4: Divide each constant term in the factored form by the original coefficient of the leading term. Original coefficient of the leading term: 10 (x + 5)(x - 6)

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**Then reduce the fractions:**

(x + 1)(x - 3)

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**Step 5: Now we use the “bottoms up” method- Move the denominator into the numerator to get:**

(2x + 1)(5x – 3) (x + 1)(x - 3)

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(2x + 1)(5x – 3) Check it- use FOIL: 10x2 -6x + 5x – 3 10x2 - x – 3

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You Try! 6x2 - 2x – 28 Step 1: multiply the constant (c) term by the coefficient (a), of the leading term Constant is -28 Leading term is 6 Therefore 6 * -28 is -168

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6x2 - 2x – 28 Step 2: Rewrite the equation by replacing the constant with the number in step 1, and remove the leading coefficient. x2 -2x -168

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**x2 -2x -168 Step 3: Factor the new equation ( x -14 )(x +12 )**

Factors of -168 Sum of factors -2, -84 no -4, 42 -8, 21 13 -12, 14 2 Switch -14, 12 -2

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( x -14 )(x +12 ) Step 4: Divide each constant term in the factored form by the original coefficient of the leading term. Original coefficient of the leading term: 6 ( x -14 )(x +12 )

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( x -14 )(x +12 ) Then reduce the fractions: ( x -7 )(x +2 ) Step 5: Now we use the “bottoms up”

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**( 3x -7 )(x +2 ) Check your answer with factoring (your choice).**

Conversation with a caution 6x2 - 2x – 28 Purpose of factoring

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**Quick steps! Step 1: multiply (c) (a),**

Step 2: Rewrite the equation by replacing the constant with the number in step 1, and remove the leading coefficient. Step 3: Factor the new equation Step 4: Divide each constant term in the factored form by the original coefficient of the leading term and then reduce the fractions: Step 5: Now we use the “bottoms up”

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Algebra 1B Chapter 9 Solving Quadratic Equations By Graphing.

Algebra 1B Chapter 9 Solving Quadratic Equations By Graphing.

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