Properties of Equations:
Given that A, B and C represent real or complex algebraic
expressions, then the following are true:
Addition Property of Equality: If A = B, then A + C = B + C.
Note that since subtracting from both sides is the same as adding a
negative amount to both sides, subtraction from both sides also is
covered by this property.
Multiplication Property of Equality: A = B, then AC = BC.
Division Property of Equality: If A = B, then A/C = B/C
where C≠0.
Absolute Value Equation Property: If
A = B, then A = B and A = B are
both possible solutions.≠0.
Simple Quadratic Equations: If A^{2} = B, and
A is unknown, then A = +/√B.
Using this property is known as Extracting Square Roots.
General Quadratic Equations: If ax^{2} + bx +
c = 0 where a, b, and c are real coefficients and x is a variable,
then . This is
known simply as The Quadratic Formula. Proof is left as
an exercise.
Radical Equations: If √A
= B, and A is unknown, then A = B^{2} is an equation
that when solved results in possible solutions of A. This is
known as Eliminating The Radical By Squaring. We could also
call this property
.
Rational Exponent Equations: If A^{M/N}
= B, and A is unknown and M is even, then A = +/B^{N/M}
is an equation that when solved results in possible solutions
of A. This is known as Eliminating a Rational Exponent. This
method also applies to solving nth root radical equations. For
example, to solve an equation with a fourth root radical, take the 4th
power of both sides.
Rational Exponent Equations: If A^{M/N}
= B, and A is unknown and M is odd, then A = B^{N/M} is
an equation that when solved results in possible solutions of
A.This is known as Eliminating a Rational Exponent. This method
also applies to solving nth root radical equations. For example, to
solve an equation with a third root radical, take the 3rd power of
both sides.
Zero Product Law: If XY = 0, then X=0 or Y=0 or both X and Y
= 0.
Note: X and Y are often factors. So for example If (x +
3)(2x 1) = 0, then we can conclude that x+3 = 0 or 2x1=0 and
we can solve each equation to find all solutions for x.
