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What are quadratic equations, and how can we solve them?

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. What is a quadratic equation?.

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What are quadratic equations, and how can we solve them?

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  1. 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

  2. What is a quadratic equation? • General form of ax2+bx+c=0, where a≠0. • An equation with one unknown in which the highest exponent is 2. • Ex. • 3x2-4x+1=0 • 10x-21=x2 • x2-4=3x

  3. 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

  4. 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. x2+5x+4=0 • What are factors of 2? • 2 and 2 Sum to 4 • 4 and 1 Sum to 5 <------- Answer • (x+4)(x+1)=0

  5. What are some clues we can use when we factor? • If c is positive, then both binomials will have the sign of b. • Ex. x2-5x+4=0 ----> (x-4)(x-1)=0 • If c is negative, one binomial will have a + and the other will have a - • Ex. x2+3x-4=0 ----> (x+4)(x-1)=0 • Ex. X2-3x-4=0 ----> (x-4)(x+1)=0

  6. 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. 2x2-7x+3=0 • (2x-1)(x-3)=0

  7. 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.

  8. Examples • x2+7x+12

  9. Special cases • x2-4=0 • x2+8x+16=0

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

  11. Review • Get into standard ax2+bx+c=0 form • Factor • Set each factor equal to zero • Solve for x • Check solutions

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

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