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/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
- 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
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/xa2 – c/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
