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What are quadratic equations, and how can we solve them? Do Now: (To turn in) What do you know about quadratic equations? Have you worked with them before? What do the graphs look like? How are they used in real life? HW: pg 76, 1-6 Do Now: (To turn in) What do you know about quadratic equations? Have you worked with them before? What do the graphs look like? How are they used in real life? HW: pg 76, 1-6

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What is a quadratic equation? General form of ax 2 +bx+c=0, where a≠0. An equation with one unknown in which the highest exponent is 2. Ex. 3x 2 -4x+1=0 10x-21=x 2 x 2 -4=3x General form of ax 2 +bx+c=0, where a≠0. An equation with one unknown in which the highest exponent is 2. Ex. 3x 2 -4x+1=0 10x-21=x 2 x 2 -4=3x

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How can we solve quadratic equations? As is the case often in math, there are many ways to solve these problems. Factoring Graphing Completing the square Quadratic Formula As is the case often in math, there are many ways to solve these problems. Factoring Graphing Completing the square Quadratic Formula

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How can we use factoring to solve a quadratic equation? When we factor, we are looking for two binomials that, when multiplied, form our quadratic. When a=1, then we are looking for factors of c that equal b when they are added together. Ex. x 2 +5x+4=0 What are factors of 2? 2 and 2Sum to 4 4 and 1Sum to 5 <------- Answer (x+4)(x+1)=0 When we factor, we are looking for two binomials that, when multiplied, form our quadratic. When a=1, then we are looking for factors of c that equal b when they are added together. Ex. x 2 +5x+4=0 What are factors of 2? 2 and 2Sum to 4 4 and 1Sum to 5 <------- Answer (x+4)(x+1)=0

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What are some clues we can use when we factor? If c is positive, then both binomials will have the sign of b. Ex. x 2 -5x+4=0 ----> (x-4)(x-1)=0 If c is negative, one binomial will have a + and the other will have a - Ex. x 2 +3x-4=0 ----> (x+4)(x-1)=0 Ex. X 2 -3x-4=0 ----> (x-4)(x+1)=0 If c is positive, then both binomials will have the sign of b. Ex. x 2 -5x+4=0 ----> (x-4)(x-1)=0 If c is negative, one binomial will have a + and the other will have a - Ex. x 2 +3x-4=0 ----> (x+4)(x-1)=0 Ex. X 2 -3x-4=0 ----> (x-4)(x+1)=0

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What happens if a≠1? If a≠1 then we need to be careful about how we set up the factors. The first terms of the binomials must multiply to form the first term of the quadratic. Ex. 2x 2 -7x+3=0 (2x-1)(x-3)=0 If a≠1 then we need to be careful about how we set up the factors. The first terms of the binomials must multiply to form the first term of the quadratic. Ex. 2x 2 -7x+3=0 (2x-1)(x-3)=0

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Ok…so what do we do with the factors? If we have two numbers multiplied together to form zero, we know that one of these numbers must be zero. (x+2)(x+3)=0 If this is true, then (x+2)=0 or (x+3)=0 Solve each problem. x=-2 or x=-3 These are the two values of x that make the equation true. If we have two numbers multiplied together to form zero, we know that one of these numbers must be zero. (x+2)(x+3)=0 If this is true, then (x+2)=0 or (x+3)=0 Solve each problem. x=-2 or x=-3 These are the two values of x that make the equation true.

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Examples x 2 +7x+12

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Special cases x 2 -4=0 x 2 +8x+16=0 x 2 -4=0 x 2 +8x+16=0

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A little more complicated 4x-5=(6/x)

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Review Get into standard ax 2 +bx+c=0 form Factor Set each factor equal to zero Solve for x Check solutions Get into standard ax 2 +bx+c=0 form Factor Set each factor equal to zero Solve for x Check solutions

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Summary Why is a quadratic equation more difficult to factor if a≠1? HW: pg 76, 1-6 Why is a quadratic equation more difficult to factor if a≠1? HW: pg 76, 1-6

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