While a polynomial of degree n has exactly n solutions (not necessarily
different or real numbers); formulae, however, have an unknown number
of forms.
One formula for solving quadratic equations is:

Where

Another formula is:

Factors

Whilst in principle all polynomials can be factorised, in practice this
might be too difficult. An easy equation to factorise is:
x^{2}-3x+2=0, which is:
(x-1)(x-2)=0
The roots of the equation are easily determined as 1 and 2.

Completing the Square

For example, lets solve the following equation:

To make a perfect square, we add half the coeffient of x (5/2) squared
to both sides:

We make a perfect square

Take the square root of both sides

x=2.5+/- 2.0616
x=4.5616, or -0.4385

Deriving the Square Roots Equation By Completing the Square

Starting with this equation:

We can convert it to normal form by dividing by a:

And add half the coefficient of x squared to both sides, because we
want to make a perfect square:

We can now make a perfect square:

Taking square roots:

And rearranging:

We finally get a familiar formula:

Eliminating the x-Term

By substituting x=t-b/2a, we can eliminate the x term in ax^{2}+bx+c,
and therefore solve the equation. We can also derive a formula with
this method.

A similar principle is used to reduce cubic equations.

Nature of the Roots

The roots of a quadratic equation are either both real or both complex,
when the coefficients are real.

Discriminant

All polynomials have discriminants, but the formulas become more and
more complex. The discriminant indicates the nature of the roots, real,
complex, etc. For the quadratic, it is:

Either the capital delta Δ, or D is used as the
symbol.
If Δ >=0, the roots are real. If it is less than zero,
then
the roots are complex. If Δ=0, then the roots are equal in
value.
In this case, each is equal to -b/4a. If the roots are equal, they are
real roots, because in an equation with real coefficients, complex
roots cannot be the same.

x^{2}-2x+1=0
The discriminant is:
Δ=b^{2}-4ac=
4-4=0
Hence the roots are equal. Each root=-b/2a=
-(-2/1)/2=1, so each root is 1

Ken's book is packed with examples and explanations that enable you to discover more than 150 techniques to speed up your arithmetic and increase your understanding of numbers. Paperback and Kindle: