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# 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 C0.

Absolute Value Equation Property: If |A| = B, then A = B and -A = B are both possible solutions.0.

Simple Quadratic Equations:  If A2 = B, and A is unknown,  then A = +/-B. Using this property is known as Extracting Square Roots.

General Quadratic Equations:  If ax2 + 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 = B2 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 AM/N  = B, and A is unknown and M is even,  then A = +/-BN/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 AM/N  = B, and A is unknown and M is odd,  then A = BN/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 2x-1=0 and we can solve each equation to find all solutions for x.

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