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

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

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

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

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

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

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.

Examples  x 2 +7x+12

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

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

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

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