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.

- Types of variation – direct, inverse, joint and partial variation
- Direction variation

Y X

Y = XK

K = ^{Y}/_{X} where K = constant

- Inverse variation

Y ^{1}/x_{2}

Y = ^{K}/x_{2}

K = yx^{2}

- Joint Variation

y x and y 2^{1}/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 = 2^{x}

^{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}

- 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.

- Factorization and Completing the square method

ax^{2} + bx + c = 0

Transferring the constant C: ax^{2} + bx = -c

Making the co-efficient of x^{2} to be 1

X^{2} + ^{b}/_{ax }= –^{c}/_{a}

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

i.e. (^{b}/_{2a})^{2} to both sides

x^{2} + ^{b}/_{ax} + (^{b}/_{2a})^{2} = –^{c}/_{a} + (^{c}/_{2a})^{2}

x^{2} + ^{b}/_{ax} + (^{b}/_{2a})^{2} = ^{b2}/_{xa2} – ^{c}/_{a}

x^{2} + ^{b}/_{ax} + (^{b}/_{2a})^{2} = __b ^{2} – 4ac__

4ac

Combining the square on the left side

(x + ^{b}/_{2a})^{2} = __b ^{2} – 4ac__

4ac

Taking square roots of both sides

X + ^{b}/_{2a} = __± ____b ^{2} – 4ac__

2a

Subtraction ^{b}/_{2a} from both side

:. X = ^{-b}/_{2a} __± ____b ^{2} – 4ac__

2a

X = __-b____b ^{2} – 4ac__

2a