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ALTERNATIVE ALGEBRAIC FORMULA FOR QUADRATIC EQUATIONS.

EBUMDON HARRISON

Mathematics Department

College of Education Agbor

ABSTRACT

Quadratic equation is an equation of the form ax2 + bx + c= 0 ie whose highest degree is two (2). This work has used quadratic equation to put forth an alternative algebraic formula for the solving of quadratic equations. The alternative formula has been shown to obtain roots of quadratic equations accurately. The formula is

X =     -2c B ±Öb2 –4ac

1.   Introduction

An equation is a mathematical statement indicating that two or more expressions are equal. An expression of the form ax2+ bx + c = 0 which is an algebraic expression of degree two is a quadratic equation where a,b and c are real constants and a ¹ 0, since when a = 0 the equation is no longer quadratic.

1.1    Method Of Solving Quadratic Equations

There are many methods of solving quadratic equations.

Among the methods are:

i.                   Factorization

ii. Completing the square

iii. Graphical method and

iv. Algebraic formula

1.2    Algebraic Formula

The quadratic formula states that for ax2+ bx + c = 0,

X = b ±Öb2 –4ac

2a

This has been in use for years.

Another algebraic formula that can solve the quadratic equation (the alternative formula) is

X =     -2c b ±Öb2 –4ac

1.2.1 Derivation Of The Formula

The general form of quadratic equation is given by ax2+ bx + c = 0

taking ax2 to the right hand side gives  bx+ c = - ax2

multiplying both sides by 4c gives (provided c ¹ 0)

4cbx + 4c2 = -4cax2

Add (xb2) to both sides, to make the left hand side of the equation a perfect square.

(xb) 2 + 4cbx + 4c2 = (xb) 2 – 4cax2

By factorization of both sides

(xb) 2 + 2cbx + 2cbx + 4c2 =x2 (b2 - 4ac).

xb (xb +2c) + 2c (xb + 2c) =x2 (b2 - 4ac).

(xb +2c) (xb + 2c) =x2 (b2 - 4ac).

(xb +2c) 2 = x2 (b2 - 4ac).

Taking the square root of both sides, we have

xb +2c = ±Ö x2 (b2 –4ac)

Collecting the terms

xb ±Ö x2 (b2 –4ac) +2c = 0

xb ±Ö x2 (b2 –4ac) = -2c

xb ± x Ö (b2 –4ac) = - 2c

x (b± x Ö (b2 –4ac) = - 2c

dividing through by

[ (b± x Ö (b2 –4ac) ] gives

x =       - 2c

b± x Ö (b2 –4ac)

This is an alternative formula for the quadratic equation

1.2.2 Some Comparison Between These Quadratic Formula Are As Follows

x= - b ± Ö (b2 –4ac)  (1) and

2a

x=         - 2c

b± Ö (b2 –4ac)    - (2)

Find the roots of the equation x2 + 4x+ 3 = 0

Solution

A= 1, b = 4, c = 3

Using equ. (1)

x = - b ±  Ö (b2 –4ac)

2a

x = - 4 ±  Ö (4)2 –4 (1) (3)

2 (1)

x = - 4 ±  Ö (16 –12)

2

x = -4 ±  Ö 4

2

x = - 4 +2 or  - 4- 2

2          2

x = -2 or   -6

2       2

x = -1 or  -3

Using equ. (2)

x=         - 2c

b± Ö (b2 –4ac)

x=         - 2 (3)

4± Ö (4)2 –4(1) (3)

-6

x=  4±  Ö (16 – 12)

x =                 -6

4± Ö 4

x=  -6   or   -6

4-2           4+2

x= -6   or -6

2        6

x = -3 or -1

Find the roots of the equation x2 – 4x + 5 = 0

Using equ. (1)

x = - b ±  Ö (b2 –4ac)

2a

a = 1, b = 4, c = 5

x = 4 ±  Ö (- 4)2 –4 (1) (5)

2 (1)

x = 4 ±  Ö (16 –20)

2

x = 4 ±  Ö - 4

2

x = 4 ±  Ö - 4

2

x = 4 ± ( Ö -1  Ö4

2

Recall from complex no.  Ö -1 = i

x = 4 ±  Ö 4. i

x = 4 ± 2i

2

x = 4+2i    or   4 – 2i

2                 2

x=2 + I or   2 – I

Using equ. (2)

x=         - 2c

b± Ö (b2 –4ac)

x=         - 2 (5)

- 4± Ö (4)2 –4(1) (5)

-10

x= - 4±  Ö (16 – 20)

-10

x =      - 4±  Ö - 4

x=                  10

- 4 ± (Ö - 4.  Ö -1 )

recall that Ö- 1 = i from complex number

-10

x =      - 4± 2i

x =   - 10       or       - 10

- 4 + 2i        - 4 – 2i

x = - 10          or      - 10

- (4 – 2i)        - (4+ 2i)

x = 10    or    10

4- 2i       4+ 2i

Multiply by the complex conjugate

x=  10 ( 4 + 2i)         or    10 ( 4- 2i)

(4 – 2i) ( 4+ 2i)         (4+ 2i) (4- 2i)

x= 40 + 20i      or   40 – 20i

16 – 4i2          16 – 4i2

recall that i2 = -1

x =    40 + 20i      or   40 – 20i

16 – (4 . 1)        16 – (4 . 1)

x =   40 + 20i      or   40 – 20i

16 – (- 4)          16 – (-4)

x =   40 + 20i      or   40 – 20i

16 + 4             16 + 4

x =   40 + 20i      or   40 – 20i

20                                                   20

x = 2+ i               or    2 - i

This can also be verified using the roots of quadratic equation formula.

Let the roots of quadratic equation be a and b.

Using formula (1)

If a = - b + Ö (b2 –4ac)

2a

and b = (- b -   Ö (b2 –4ac)

2a

let Ö b2 –4ac =  ÖD

a + b = - b + ÖD + ( -b - ÖD )

2a                2a

a + b = - b + ÖD – b - ÖD

2a

= 2b = -b

2a   a

Using the new formula

a =        -2c

b + Ö b2 –4ac

and

-2c

b =    b - Ö b2 –4ac

let Öb2 –4ac = ÖD

a+ b = -2c         +   - 2c

(b + ÖD)       (b - ÖD)

= - 2c (b-  ÖD) - 2c ( – b - ÖD)

b2 - (D)

a + b         =   - 4cb

b2 - (D)

recall that D = Ö b2- 4ac

a + b = - 4cb                  =  - 4cb

b2 - (b2 - 4ac)       b2 – b2 + 4ac

a + b = - 4cb =     - b

4ac           a

Obviously, the results obtained when we use the original formula (1)

x = - b ±  Ö (b2 –4ac)   and the alternative formula

2a

x=         - 2c             are the same

b± Ö (b2 –4ac)

Conclusion This formula X =         - 2c

b ±  Ö (b2 –4ac)

has added to the existing algebraic formulae. As the saying goes, “ the more the merrier”. It may afford students an opportunity to recall/ use an alternate formula. This formula like the original solves all quadratic equations of any form and size.

If b2 –4ac is positive, the roots are real and unequal; if b2 –4ac is equal to zero the roots are identical,  b2 –4ac is negative, then the roots are imaginary.

References:

eBeyer, H.W (1985) CRC Standard Mathematics Tables, 26th Edition Florida

CRC Press.

Erwin, K. (2008) Advanced Engineering Mathematics, Ohio John Wily and

sons Inc.

Macral M. F Kalejaiye A.O et al (2001) New General Mathematics for

Senior Secondary Schools 3rd edition, Lagos Academy press.

Okonta P.N (2005) Introductory Pure mathematics, Lagos Functional

Publishing Company.

Stroud K.A and Dexter J. (2007). Engineering Mathematics sixth edition

New York Palgrare Macmillan

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