Solving equations

"A demonstration is not anything other than the resolution of a truth in other truths already known."
Gottfried Wilhelm Leibniz (1646 - 1716)
German mathematician, philosopher, scientist, diplomat, librarian and lawyer

Solving an equation, for example where is an algebraic expression containing the unknown , is to find all solutions of the equation, that is to find all the values of the number such that equality is true.

Example: For the equation  , We can verify that    is a solution.
Indeed, if we replace by , we actually get

Thus, is a solution of this equation. Nevertheless, can not claim to have solved it because we do not know a priori if there are others.
We do not know all solutions.
We could also check that is an other solution:

So we know a second solution, but again we can not yet claim to have solved the equation ...

The purpose of what follows is just solving equations, that is to say, the determination of all solutions of an equation (to find them, and be sure to get all of them).

One can solve only five types of equation. All other equations (second or higher degree equations, trigonometric equations, logarithmic, ...), are then based on these five types.

  1. Linear equations (first degree equations):  

    solved according to:      .

  2. Zero factor equations:   

    solved using the "zero factor property" (or "Rule" or "Principle"): if a product of two terms is zero then at least one of the terms has to be zero

    Note 1: Of course, there may be more than two factors, eg for three factors:

    Note 2: These equations are fundamental. They allow to decompose, equivalently, an equation into several simpler equations.
    An equation may not directly and apparently be displayed as products of factors; it is often possible to transform to get a factored expression.
    For this particular reason, factorization is a fundamental operation in mathematics.

  3. Zero quotient equations:   

    which are very similar to zero factor equations, except that here the denominator can not be zero:

    Note: Values of for which denominateur is zero: , quite apart from any equation, are such that the quotient does not exist (division by does not exists !!).
    These values of ar prohibited values for the expression and cannot, in any way, be solutions.

  4. (Perfect) square equations:   

    for which, depending on the sign of the real number :
    • if :   then equation  has no solution;
      (because a square can not be equal to a negative number)
    • if :   then  if and only if .
    • if :  then   if and only if

  5. Square root equations:  

    for which, depending on the sign of ,

    • if :   then equation  has no solution
      (a square root can not be a negative number)
    • if :   then   if and only if  and  .

    Note: Valeurs of for which , quite appart from any equation, are such that the quotient, are such that the square root does not exist (the square root of a negative number does not exist for real number !! ).
    These values are prohibited values for the expression and cannot, in any way, be solutions.

We now give an example for each type of equation.

Example 1:

is a linear (or first degree) equation:

Example 2:

is a zero factor equation, thus

These two last equations are now simpler linear equations

Finally equation has two solutions: and .

Example 3:

is a zero quotient equation, thus:

These two last equations can then simply be solved

is the solution of the equation , because we check that  (  is a prohibited value for the quotient).
Equation has thus a unique solution .

Example 4:

is a (perfect) square equation  , with  , thus :

We can then solve these two last equations:

Thus, equation has two solutions and .

Example 5:

is a square root equation: , with :

First equation is a linear one and can simply be solved:

We moreover check that for ,   we have .
Equation thus has a unique solution .

Exercices Solve the following equations: