:: Equations with rational expression - Solving rational equations

  To solve an equation with rational expressions (fractions), determine the lowest common denominator (LCD) of all rational expressions in the equation and multiply each term of both sides of the equation by the common denominator to eliminate fractions. Then, solve the equation that remains.

Note, check for extraneous solutions. The extraneous solutions are values that cause any denominator in the equation to be 0. So, these values have to be excluded from the solution.
Rational equations - linear equations
Example:  Solve rational linear equations,
Rational equations - quadratic equations
Example:  Solve the rational quadratic equation,
 :: Binomial equations
An equation of the form,      axn b = 0,   
a > 0,  b > 0 and n is a natural number is called the binomial equation.
Solving binomial equations
Using substitution the equation transforms to or y n  1 = 0.
then we apply the binomial factorization

Example:  Solve the binomial equation  8x3 - 27  = 0.

Solution:   Let substitute

To obtain the values of the original variable plug the solutions into the substitution  x = (3/2)y

 :: Equations reducible to quadratic form, bi-quadratic equations
A bi-quadratic equation is said to be reducible to quadratic if the variable factor of the leading term is the square of the variable factor in the middle term.
Solving bi-quadratic equations or equations reducible to quadratic
Therefore, a bi-quadratic equation
ax4 + bx2 + c = 0   we can write   a(x2)2 + bx2 + c = 0
and solve as the quadratic equation in the unknown x2 using the substitution  x2 = y.

Example:  Solve the bi-quadratic equation  3x4 - 4x2 + 1 = 0.

Solution:   By substituting  x2 = y  we get the quadratic equation
3y2 - 4y + 1 = 0

To obtain the values of the original variable plug the solutions into the substitution  x2 = y

 :: Radical or irrational equations
Any equation where the variable is inside a radical is called an irrational equation.
To solve an irrational equation we should isolate one of the radicals on one side of the equation and get other radicals and terms on the other side of the equation.
Rise both sides of the equation to a power to remove the radical, and then simplify and solve as we do with any equations. Repeat the procedure until all radicals are removed.
Substitute answers back into original equation to make sure that solutions are valid, as there could be some additional or extraneous solutions that do not satisfy the original equation.
Solving irrational or radical equations
Example:  Solve the irrational equation
Solution:   First isolate the radical and then square both sides of the equation,

Checking for the solutions,

Example:  Solve the irrational equation

Checking for the solutions,

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