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QUADRATIC EQUATIONS

CONTENT

  • Construction of Quadratic Equations from Sum and Product of Roots.
  • Word Problem Leading to Quadratic Equations.

 

CONSTRUCTION OF QUADRATIC EQUATIONS FROM SUM AND PRODUCT OF ROOTS

We can find the sum and product of the roots directly from the coefficient in the equation. It is usual to call the roots of the equation α and β  If the equation

ax2 +bx + C = 0   …………….  I

has the roots α and β then it is equivalent to the equation

(x – α  )( x – β )  = 0

x2 – βx –  βx + αβ  = 0 ………… 2

Divide equation (i)by the coefficient  of x2

 ax2+ bx +    = 0  …………   3

aaa

Comparing equations (2) and (3)

x2 +  b x  +  C     = 0

aa

x2  –  ( α +β)x  + αβ    = 0

then

α+ β= -b

a

and αβ = C

a

For any quadratic equation, ax2 +bx + C = 0 with roots α and β

α + β  = –b

a

αβ =  C

a

 

Examples

  1. If the roots of 3x2 – 4x – 1 = 0 are αand β, find α + β and αβ
  2. if α and βare the roots of the equation

3x2 – 4x – 1 = 0 , find the value of

(a)     α     +  β

β          α

(b)   α   –   β

Solutions

  1. Since α + β = -b

a

Comparing the given equation 3x2 – 4x – 1= 0 with the general form

ax2 + bx + C = 0

a = 3, b = -4, C = 1.

 

Then

α + β = –-(-4)

a       3

= + =  +1 1/3

3

αβ =C   =  –1    = -1

a          3         3

2.aα +  β    =  α22

β        α    αβ

= (α + β )2 – 2αβ

αβ

Here, comparing the given equation, with the general equation,

a = 3, b = -4, C = – 1

from the solution of example 1 (since the given equation are the same ),

α + β = –b =  – (-4)   = +4

3           3

αβ  = = – 1

a        3

then

α  +    β    =  ( α+ β ) 2 – 2 αβ

β         α                  αβ

=  (4/3 ).2 – 2 ( – 1/3 )

  • 1/3

 16        ± 2

   9           3

 – 1

3

 16 + 6   ÷    -1/3

9

22     x   -3

9          1

=   -22

3

or  α + β     = – 22   = – 7 1/3

β        α           3

  1. b) Since

(α-β) 22 + β– 2 α β

but

α2 + β2 = ( α + β)2 -2 α β

:.(α- β)2 = ( α+ β )2 – 2αβ -2αβ

(α – β)2 = (α + β )2 – 4α β

 

:.( α – β) = √(α + β )2 – 4αβ

( α – β) =√   (4/3 )2 – 4 ( – 1/3 )

= √ 16/9  +4/3

=    √16 + 12

9

=      √28           =   √28

9                    3

:. α – β  =         √28

3

Evaluation

If α and β are the roots of the equation

2x2 – 11x + 5 = 0, find the value of

  1. α – β
  2. 1 +        1

α + 1             β+ 1

  1. α2+ β2

 

WORD PROBLEM LEADING TO QUADRATIC EQUATIONS

Examples

  1. Find two numbers whose difference is 5 and whose product is 266.

Solution

Let the smaller number be x.

Then the smaller number be x+5.

Their product is x(x+5) .

 

Hence,

x(x+5) = 266

x2+5x-  266 = 0

(x-14)(x+19)=0

x=14 or x= -19

 

The other number is 14+5 or  -19+5 i.e 19 or -14

:. The two numbers are 14 and 19 or -14 and -14.

 

  1. Tina is 3 times older than her daughter. In four years time, the product of their ages will be 1536. How old are they now?

 

Solution

Let the daughter’s age be x.

Tina’s age = 3x

In four years’ time,

Daughter’s age = (x+4)years

Tina’s age = (3x+4)years

The product of their ages :

(x+4)(3x+4)= 1536

3x2+ 16x – 1520 = 0

(x-20)(3x+76) = =0

x=20 or x=-25.3

 

Since age cannot be negative, x=20years.

:. Daughter’s age = 20years.

Tina’s age = 20×3=60years.

 

Evaluation

  1. Think of a number, square it, add 2 times the original number. The result is 80. Find the number.
  2. The area of a square is 144cm2 and one of its sides is (x+2)cm. Find x and then the side of the square.
  3. Find two consecutive odd numbers whose product is 224.

 

GENERAL EVALUATION/REVISION   QUESTIONS

  1. The area of a rectangle is 60cm2. The length is 11cm more than the width. Find the width.
  2. A man is 37years old and his child is 8. How many years ago was the product of their ages 96?
  3. If α and β are the roots of the equation 2x2 – 9x+4=0, find
  4. a) α + β (b) αβ (c) α – β   (d)  αβ/ α + β

 

WEEKEND ASSIGNMENT

If α and β are the roots of the equation 2x2 + 9x+9=0:

  1. Find the product of their roots. A. 4 B. 4.5   C. 5.5    B. -4.5
  2. Find the sum of their roots. A. 4 B. 4.5   C. 5.5    B. -4.5
  3. Find α22 A. 11.5 B. -11.25   C. 11.25   D. -11.5
  4. Find αβ/ α + β                               A. 1  B.-1    C.  1.5     D. 4.5

 

THEORY

  1. The base of a triangle is 3cm longer than its corresponding height. If the area is 44cm2, find the length of its base.
  2. Find the equation in the form ax2+bx+c=0 whose sum and products of roots are respectively:
  3. a) 3,4 (b)  -7/3  , 0    (c) 1.2,0.8

 

See also

GEOMETRIC PROGRESSION

ARITHMETIC PROGRESSION (A. P)

PERCENTAGE ERROR

LOGARITHM OF NUMBERS

MENSURATION

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