Simple and Quadratic Equation

Simple and quadratic equation refers to a mathematical equation that expresses the relationship between the two expressions on both sides of the ‘equal’ symbol. Such equations consist of squares and variable, usually in the form of x or y.

 

  1. Types of variation – direct, inverse, joint and partial variation
  2. Direction variation

Y  X

Y = XK

K = Y/X where K = constant

 

 

 

  1. Inverse variation

Y  1/x2

Y = K/x2

K = yx2

 

 

 

  • Joint Variation

y  x and y 21/p2

y  x  1/p2

y x/p2

y = dk/p2

yp2 = dk

k = yp2/d

 

 

 

Example:  If y varies directly as d and inversely as the square of x.

If d = 30, x = 24 and y = 1/18,

Find the law connecting the variables. Also find y when d = 5 and                     x = 2

 

 

Solution:

y  d and y  1/x2

y  d  1/x2

y  d/x2

y = dk/x2

 

 

When y = 1/18, d = 30, x = 2x

1/18 = 30k/2×2

1/18 = 30k/576

k = 1/18 x 576/30

k = 16/15

 

:- the law connecting the variable will be

y = 16d/15×2

 

To find y when d = 5, x = 2

y = 16×5/15×22

16×5/16×4 = 4/3

 

 

 

  1. Partial variation

e.g. d = a, d  f

d = a + dk (where a and k are constant)

 

QUADRATIC EQUATION

 

The definition of a quadratic equation is any equation contains a term, where the unknown is squared, and there is no term, where it is promoted to a higher power.

 

  1. Factorization and Completing the square method

 

ax2 + bx + c = 0

Transferring the constant C: ax2 + bx = -c

Making the co-efficient of x2 to be 1

X2 + b/ax = –c/a

Then add the square of half the co-efficient of x

i.e. (b/2a)2 to both sides

x2 + b/ax + (b/2a)2 = –c/a + (c/2a)2

x2 + b/ax + (b/2a)2 = b2/xa2c/a

x2 + b/ax + (b/2a)2 = b2 – 4ac

4ac

 

Combining the square on the left side

(x + b/2a)2 = b2 – 4ac

4ac

 

Taking square roots of both sides

X + b/2a = ± b2 – 4ac

2a

 

Subtraction b/2a from both side

:.  X = -b/2a ± b2 – 4ac

2a

X = -bb2 – 4ac

2a

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