Topic 6Applications of Quadratic Equations

Unit 7 Topic 6

Information

A quadratic equation can be solved in a variety of ways.

Factoring• Some quadratic equations can be solved by

factoring.• To factor an equation, start by writing the

equation in standard form.• Factor using the common factor, a difference of

squares or a trinomial method. • Set each factor equal to zero and solve the

resulting linear equations. Each root is a solution to the original equation.

Quadratic Formula•The roots of a quadratic equation in standard form, , where a 0, can be determined by using the quadratic form .• A quadratic equation can have one distinct real root, two distinct real roots or no real roots.

Graphing•Method 1: Find the x-Intercepts of the Corresponding FunctionYou can solve quadratic equations of the form , by graphing the corresponding quadratic function, , and finding the x-intercepts of the graph of .

20 ax bx c

21y ax bx c 1y

• Method 2: Find the x-Coordinates at the Points of Intersection of a System of EquationsYou can solve quadratic equations of the form , by graphing the corresponding quadratic functions, and , and then finding the x-coordinate at each point of intersection of and .

2 2ax bx c dx ex f 2

1y ax bx c 2

2y dx ex f1y 2y

When you use a quadratic equation to solve a problem, you need to verify that each of the roots make sense in the context of the situation.

Example 1Solving a Problem Using Factoring

The manager of a fashion store is investigating how raising or lowering the price of a purse changes the daily revenue from the purse. The function gives the store’s revenue, R, in dollars, from purse sales, where x is the price change, in dollars. a) Write a quadratic equation that you could use to find what price changes will result in no revenue?

Try this on your own first!!!!Try this on your own first!!!!

2 15 100R x x

Example 1a: SolutionWrite a quadratic equation that you could use to find what price changes will result in no revenue?

2

20

1

1

5 100

5 100x x

R x x

Example 1Solving a Problem Using Factoring

b) Use factoring to determine what price changes will result in no revenue.

2

20 ( 15 100)

0 1( 5)( 20)

5, 20

11 1 1

1

5 1000

x x

x

x

x

x

x GCF of -1

Product = -100

Sum = -155, -20

A decrease by $5 or an increase by $20 will result in no revenue.

Example 2Solving a Problem Using the Quadratic Formula

Eric makes an open-topped box from a piece of cardboard measuring 14 cm by 10 cm. He forms the sides of the box by cutting out squares of x centimetres from each corner of the rectangular sheet, as shown in the diagram below.

Try this on your own first!!!!

Example 2

Eric wants the area of the base of the box to be 45 cm2. Eric lets x represent the side length of the square cut from each corner. Into the area formula for a rectangle, , Eric substituted A = 45, and .

Finish Eric’s solution.

a) Express the equation in standard form, .

Try this on your own first!!!!

45 14 2 10 2

A lw

x x

10 2w x 14 2l x

2 0ax bx c

45 14 2 10 2

A lw

x x

A lw

Example 2a: Solution

Express the equation in standard form, .

2 0ax bx c

2

2

220 4

45 4 4

45 14 2 10

8

14

140

0 4 4

2

4

8

0

95

85 2

x x

x

x

x

x

x

x x

Example 2b: SolutionUse the quadratic formula to solve for x.

2

2

248 48 4 95

4

48 2304 15208

48 7848

48 28

0 4 48 95

42

4

2

8

x x

b b acx

a

x

x

x

x

48 28 76

9.58 8

x

48 28 20

2.58 8

x

You have to enter this in your calculator twice. Once with the positive and once with the negative.

a b c a = +4b = -48c = +95

Example 2c: SolutionWhat are the dimensions of the box?

10 2w x

14 2 9.5

2

5

14l

l

x

l

9.5x 2.5x

14 2 2.5

9

14 2l

l

l

x

10 2 2.5

5

10 2w

w

w

x

It is not possible to cut out 9.5 cm from each end of a 14 cm sheet.

The only possible dimensions of the box, such that the area of the base is 45 cm2, are a length of 9 cm, a width of 5 cm, and a height of 2.5 cm.

Cut out squares with side length of 2.5 cm (equal to the height of the box) in each corner of the cardboard.

Example 3Solving a Problem Using Graphing

A football kicker kicks or punts a football into the air. The path of the ball can be modelled by the function ,where h is the height of the football, in feet, and d is the ball’s horizontal distance, in feet, from the point of impact with the kicker’s foot.

Try this on your own first!!!!

20.01 1.1 2h d d

d

h

Example 3Solving a Problem Using Graphing

a) Suppose the nearest defensive player jumps up to a height of 9 feet and attempts to block the punt, as shown in the diagram above. Graphically determine how far the defensive player is from the point of impact. Answer to the nearest tenth of a foot.

Try this on your own first!!!!

Example 3a: SolutionSuppose the nearest defensive player jumps up to a height of 9 feet and attempts to block the punt, as shown in the diagram above. Graphically determine how far the defensive player is from the point of impact. Answer to the nearest tenth of a foot.

2

1

2

22

9 0.01 1.1 2

9

0.01 1.1 2

6.8

0.01 1

fee

. 2

t

1

x x

y

y x x

x

h d d

2nd trace5: intersect

Horizontal distance (ft)x

yH

eig

ht

(ft)

The defensive player is 6.8 feet from the point of impact.

Possible Window Settings

Example 3b: SolutionSuppose the ball is not blocked by the defensive player or any other player. Graphically determine how far down the field the ball will go when it hits the ground, rounded to the nearest tenth of a foot.

2

2

0 0.01

0.

1.1 2

01 1.1

111.8 fe

2

et

h x x

x x

x

0h

2nd trace2: zero

The ball will have travelled about 111.8 feet when it hits the ground.

Possible Window Settings

Need to Know:• There are many real-life applications that can be

represented by quadratic equations.• A quadratic equation can be solved in a variety of

ways, such as; graphically or using the quadratic formula. Some quadratic equations can be solved by factoring.

• When you solve a quadratic equation, verify that the roots make sense in the context of the problem.

You’re ready! Try the homework from this section.