**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

ax^{2} +bx + C = 0 ……………. I

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

(x – α )( x – β ) = 0

x^{2} – βx – βx + αβ = 0 ………… 2

Divide equation (i)by the coefficient of x^{2}

__ ax ^{2}__+

__bx__+

__C__= 0 ………… 3

aaa

Comparing equations (2) and (3)

x^{2} + __ b x __ + __ C __ = 0

aa

x^{2} – ( α +β)x + αβ = 0

then

α+ β= __-b__

a

and αβ = __C__

a

For any quadratic equation, ax^{2} +bx + C = 0 with roots α and β

α + β = –__b__

a

αβ = __ C__

a

**Examples**

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

3x^{2} – 4x – 1 = 0 , find the **value** of

(a) __α__ + __ β__

β α

(b) α – β

**Solutions**

- Since α + β =
__-b__

a

Comparing the given equation 3x^{2} – 4x – 1= 0 with the general form

ax^{2} + bx + C = 0

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

Then

α + β = –__b __= __-(-4)__

a 3

= + __4 __= +1 ^{1}/_{3}

3

αβ =__C __ = –__1 __ = __-1__

a 3 3

2.a__α __+ __ β __ = __α ^{2} +β^{2}__

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 =__ C __ = – __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

- b) Since

(α-β) ^{2} =α^{2} + β^{– 2} α β

but

α^{2} + β^{2} = ( 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

2x^{2} – 11x + 5 = 0, find the **value** of

- α – β
- 1 + 1

α + 1 β+ 1

- α
^{2}+ β^{2}

** **

**WORD PROBLEM LEADING TO QUADRATIC EQUATIONS**

**Examples**

- 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

x^{2}+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.

- 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

3x^{2}+ 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**

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

**GENERAL EVALUATION/REVISION QUESTIONS**

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

** **

**WEEKEND ASSIGNMENT**

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

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

** **

**THEORY**

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

See also