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

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?

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=010x-21=x2

x2-4=3x

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=010x-21=x2

x2-4=3x

How can we solve quadratic equations?

How can we solve quadratic equations?

As is the case often in math, there are many ways to solve these problems.FactoringGraphingCompleting the squareQuadratic Formula

As is the case often in math, there are many ways to solve these problems.FactoringGraphingCompleting the squareQuadratic Formula

How can we use factoring to solve a quadratic

equation?

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=0What are factors of 2?

2 and 2 Sum to 44 and 1 Sum 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. x2+5x+4=0What are factors of 2?

2 and 2 Sum to 44 and 1 Sum to 5 <------- Answer

(x+4)(x+1)=0

What are some clues we can use when we factor?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)=0Ex. X2-3x-4=0 ----> (x-4)(x+1)=0

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)=0Ex. X2-3x-4=0 ----> (x-4)(x+1)=0

What happens if a≠1?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

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

Ok…so what do we do with the factors?

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)=0If this is true, then (x+2)=0 or (x+3)=0Solve each problem.

x=-2 or x=-3These 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)=0If this is true, then (x+2)=0 or (x+3)=0Solve each problem.

x=-2 or x=-3These are the two values of x that make

the equation true.

ExamplesExamples

x2+7x+12x2+7x+12

Special casesSpecial cases

x2-4=0

x2+8x+16=0

x2-4=0

x2+8x+16=0

A little more complicatedA little more complicated

4x-5=(6/x)4x-5=(6/x)

ReviewReview

Get into standard ax2+bx+c=0 form

FactorSet each factor equal to zeroSolve for xCheck solutions

Get into standard ax2+bx+c=0 form

FactorSet each factor equal to zeroSolve for xCheck solutions

SummarySummary

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