5. QUADRATIC EQUATIONS

What do we learn in this module ?

• What are Quadratic Equations ?• Standard form of Quadratic Equations• Discriminants and their roots• Why Quadratic Equations ?

Introduction to Quadratic Equations

• First degree equations have variables raised to the power of 1 (one degree), as shown in the graph, and have “only one root”

Examples of first degree equations

Ex. Perimeter of a squareIf x is the side of a square,and if the perimeter is 16 units,

Perimeter = 4 . x 16 = 4 . x

x = 4 units

• Area of a square Area = x2

Area = 4 * 4 = 16 sq.units

Examples of quadratic equations

Example of solving quadratic equations :

Definition of a Standard Quadratic Equation

Standard form of a Quadratic Equation

Derivation of the standard equation

This is called Sridhara’s method

Examples of solving using standard equation

Reducing to Quadratic form

x4 − 16x2 − 225 = 0x2 = tt2 – 16t – 225 = 0

Discriminant of a Quadratic Equation

is called a discriminant

>0, there are 2 unequal real solutions.

=0, there is a repeated real solution.

<0, there is no real solution.

The sum of the squares of 2 consecutive positive even numbers is 580. Find the numbers Identify the unknown: Let one number be x, therefore 2nd number is x + 2

Form the equation2 2( 2) 580x x

Solve!

Are both answers acceptable?

(rej)

2 2

2

2

4 4 580

2 4 576 0

2 288 0

( 16)( 18) 0

16 or 18

x x x

x x

x x

x x

x

:The numbers are 16 and 18 Ans

Statement Problems

The length and breadth of a rectangle are (3x + 1) and (2x – 1) cm respectively. If the area of the rectangle is 144 cm2, find x.

Identify the unknown!

(3 1)(2 1) 144x x Form the equation!

2

2

6 3 2 1 144

6 145 0

(6 29)( 5) 0

295 or

6

x x x

x x

x x

x

Solve!

Are both answers acceptable? : 5 Ans x

(rej)

The sum of the squares of 2 consecutive positive even numbers is 580. Find the numbers .Identify the unknown: Let one number be x, therefore 2nd number is x + 2

Form the equation2 2( 2) 580x x

Solve!

Are both answers acceptable?

(rej)

2 2

2

2

4 4 580

2 4 576 0

2 288 0

( 16)( 18) 0

16 or 18

x x x

x x

x x

x x

x

:The numbers are 16 and 18 Ans

The perimeter of a rectangle is 44 cm. The area of the rectangle is 117 cm2. Find the length of the shorter side of the rectangle.

Let one side be x, therefore other side is (44 − 2x) ÷ 2 = 22 – x

(22 ) 117x x

Are both answers acceptable?

(rej)

: The shorter side is 9 cm Ans

xx

2

2

22 117

22 117 0

( 9)( 13) 0

9 or 13

x x

x x

x x

x

A rectangular swimming pool measures 25 m by 6 m. It is surrounded by a path of uniform width. If the area of the path is 102 m2, find the width of the path.

Let the width be x. Therefore, length of path = 25 + 2x, breadth of path = 6 + 2x

(25 2 )(6 2 ) 252x x 2

2

2

150 50 12 4 252

4 62 102 0

2 31 51 0

( 17)(2 3) 0

1.5 or -17(rej)

x x x

x x

x x

x x

x

25 m6 m

25 + 2x

6 + 2x

Area of pool = 25 x 6 = 150 m2

Ans: The width of the path is 1.5 m

A duck dives under water and its path is described by the quadratic function y = 2x2 -4x, where y represents the position of the duck in metres and x represents the time in seconds.

a. How long was the duck underwater?a. How long was the duck underwater?

The duck is no longer underwater when the depth is 0. We can plug in y= 0 and solve for x.

)4(20

420 2

xx

xx

x20 40 x

So x = 0 or 4

The duck was underwater for 4 seconds

The duck was underwater for 4 seconds

A duck dives under water and its path is described by the quadratic function y = 2x2 -4x, where y represents the position of the duck in metres from the water and x represents the time in seconds.

b. When was the duck at a depth of 5m?b. When was the duck at a depth of 5m?We can plug in y= -5 and solve for x.

4

244

4

40164

)2(2

)5)(2(4)4()4(

2

4

2

2

x

x

x

a

acbbx

We cannot solve this because there’s a negative number under the square root.

We conclude that the duck is never 5m below the water.

5420

4252

2

xx

xx

A duck dives under water and its path is described by the quadratic function y = 2x2 -4x, where y represents the position of the duck in metres from the water and x represents the time in seconds.

b. When was the duck at a depth of 5m?b. When was the duck at a depth of 5m?We can check this by finding the minimum value of y.

2

)1(4)1(2

14

4

)2(2

)4(2

2

y

y

x

x

x

a

bx

We conclude that the duck is never 5m below the water.

A duck dives under water and its path is described by the quadratic function y = 2x2 -4x, where y represents the position of the duck in metres and x represents the time in seconds.

c. How long was the duck at least 0.5m below the water’s surface?c. How long was the duck at least 0.5m below the water’s surface?

We can plug in y= -0.5 and solve for x.

The duck was 0.5m below at t = 0.14s and at t = 1.87s

This will give us the times when the duck is at 0.5 m below.

This will give us the times when the duck is at 0.5 m below.

5.0420

425.02

2

xx

xx

sorx

x

x

x

x

a

acbbx

87.114.0

4

46.344

124

4

4164

)2(2

)5.0)(2(4)4()4(

2

4

2

2

Therefore it was below 0.5m for 1.73s

Example f(x) = x2 - 4

4

2

-2

-4

-5 5

Solutions are -2 and 2.

Solving using Graphical method

f(x) = 2x - x2

Solutions are 0 and 2.

4

2

-2

-4

5

One method of graphing uses a table with

arbitrary

x-values.Graph y = x2 - 4x

Roots 0 and 4 , Vertex (2, -4) , Axis of Symmetry x = 2

x y0 01 -32 -43 -34 0

4

2

-2

-4

5

Why Quadratic Equations ??

http://www.youtube.com/watch?v=BjbyqgUEbAE

Balls, Arrows, Missiles and Stones If you throw a ball (or shoot an arrow,fire a missile or throw a stone) it will go up into the air, slowing down as it goes, then come down again ... and a Quadratic Equation tells you where it will be!

Quadratic Equations are useful in many other areas: Quadratic equations are also needed when studying lenses and curved mirrors.And many questions involving time, distance and speed need quadratic equations.I am pretty sure that economists need to use quadratic equations, too!