Square Roots

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:
x²-3x+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 ax²+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.

x²-2x+1=0
The discriminant is:
Δ=b²-4ac=
4-4=0
Hence the roots are equal. Each root=-b/2a=
-(-2/1)/2=1, so each root is 1

Vieta's Roots Formalas

A quadratic equation with roots α and β can be written:
ax²+bx+c=a(x-α)(x-β)=0
For which, Vieta's root forumula is:


Ken Ward's Mathematical Pages

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Faster Arithmetic - by Ken Ward