

ALGEBRA


Equations

::
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,




::
Equations
reducible to quadratic form, biquadratic equations

A biquadratic 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 biquadratic
equations or equations reducible to quadratic

Therefore, a biquadratic equation 
ax^{4}
+
bx^{2}
+
c
=
0 we can
write a(x^{2})^{2}
+
bx^{2}
+
c
=
0

and
solve as the quadratic equation in the unknown
x^{2}
using the substitution x^{2}
=
y. 

Example:
Solve the
biquadratic
equation 3x^{4}

4x^{2}
+ 1
=
0. 
Solution:
By substituting x^{2}
=
y we get the quadratic equation

3y^{2}

4y
+ 1
=
0


To obtain the values of the
original variable plug the solutions into the substitution
x^{2}
=
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 



Checking
for the solutions,

















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