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Literal equations


The 2nd degree equations in the variable x which have some coefficients or some independent terms indicated by other letters are called literal equations.

Letters that appear in a literal equation, excluding the unknown, are called parameters.

Examples:

ax2+ bx + c = 0 unknown: x parameters: a, b, c

ax2 - (2a + 1) x + 5 = 0 unknown: x parameter: a

Incomplete Literal Equations

Solving incomplete literal equations follows the same process as numerical equations. Take a look at the examples:

  • Solve the incomplete literal equation 3x2 - 12m2= 0, where x is the variable.
    Solution:
    3x2 - 12m2 = 0
    3x2 = 12m2
    x2 = 4m2

    x =
    So we have:

  • Solve the incomplete literal equation my2- 2aby = 0, with m0, being y the variable.
    Solution
    my2 - 2aby = 0
    y (my - 2ab) = 0
    We therefore have two solutions:
    y = 0
    or
    my - 2ab = 0 my = 2ab y =
    Like this:

In the solution of the last example, we would have committed a big mistake if we had thus resolved:

my2 - 2aby = 0

my2 = 2aby

my = 2ab

That way we would get only the solution .

The zero of the other solution was "lost" when we divided both terms by y.

This is a good reason to be very careful about cancellations, thus avoiding division by zero, which is absurd.

Complete literal equations

The complete literal equations can also be solved by Bhaskara's formula. Follow the example:

  • Solve the equation: x2 - 2abx - 3a2B2, where x is the variable.
    Solution:
    We have a = 1, b = -2ab and c = -3a2B2




    Therefore:

    Thus we have: V = {- ab, 3ab}.
Next: Relations between Coefficients and Roots