ALGEBRAIC EQUATIONS

Welcome to my Mathematics Blog.

Quadratic Equations

The equation of the fom
ax+bx+c = 0, a#0

is called a quadratic equation in a variable x. The vaues of x which satisfy
the quadratic equation are called its soiutions or roots.

Soution by Factorization
A method of finding the solutions of a quadratic equation
ax +bx +c= 0, a#0
is to find the linear factors of the quadratic expression (or function)
f(x) = ax^2+ bx +c, a #0
as fx) = glx). h(x) so that the quadratic equation becomes

g(x). h(x)= 0.

Then one or both of the factors must be equal to zero, i.e.
either g(x) = 0 or h(x) = 0.

This method of factorization gives the two solutions of the quadratic equation.
One way of factorising a quadratic expression ax + bx +c (where a,
and c are rational numbers) as a product of two linear factors with ratior
coefficients is to express a. c as a product of 2 factors obtained to break
the middle term, bx, into 2 terms. The quadratic function then becomes
an expression containing four terms which can then be factonized by the
grouping method.
This method of factorization can always be used if the expression b^2-4ac
known as the discriminant of the quadratic expression ax + be t C
perfect sq uare.

Example1

2x^2+5x=0

X(2x + 5)=0

= 0 or 2x +5 0=0

Check that these solutions satisfy the original equation.

Example2

y-9=0
(y+3) (y-3) = 0 (Difference of 2 squares)
yt3=0 or y-3 = 0
y-3 or y = 3

Check that these solutions satisfy the original equation.

Example 3

3p-P4 0

Using the general form of the coetficients
a=3, b= -1, c=-4.

Then the discriminant