Solving Quadratic (2nd degree) Equations

There are 3 widely used methods for solving quadratic
equations.
A quadratic (or second-degree) equation is an equation in which the
variable has an exponent of 2. The standard form of a quadratic
equation is _{}.

The three methods used to solve
quadratic equations are:
1)
factoring, 2) the square root property, and 3) the quadratic formula.
Quadratic equations generally have 2 solutions.

1) *Factoring* is one method used
to solve second-degree and larger equations.
First the equation must be written in standard form. This means that the
polynomial must be in descending form and set equal to zero. Next, you
must factor the polynomial. (You may want to review
factoring.) Once the polynomial is factored, set any
factor which contains a variable equal to zero and solve (using isolation)
for the variable. Check your answer in the original equation. Every
quadratic equation has two solutions, although in the case of a perfect
square trinomial both of the solutions are the same.

If the equation is factored
and set equal to zero as in the example,
, then and .

Solve the
equation, . First, you must get the equation in standard form.

This means that you must add 16 to both sides so that the 16 is removed
from the left side of the equation and the equation will then be equal to
zero
. Now you are ready to factor,
. Set the factors equal to zero. Since
is repeated twice as a factor, there are two solutions,
but they are both the same. Thus,
is the only unique solution to this problem. This is
a perfect square trinomial, which factored into the square of a binomial.

If the problem has a degree
of three (in other words the variable in the equation is cubed) then you
will find three solutions. Example:
.

First, factor the GCF of +2x from each of the terms of the polynomial
.

Next factor, . We now have three factors which contain variables,
therefore, and and . This example of a third degree equation has three
solutions .

2) The __square root
property__ involves taking the square roots of both sides of an
equation. Before taking the square root of each side, you must isolate
the term that contains the squared variable. Once this squared-variable
term is fully isolated, you will take the square root of both sides and
solve for the variable. We now introduce the possibility of two roots for
every square root, one positive and one negative. Place a _{} sign in front of the side containing the constant
before you take the square root of that side.

Example 1:

_{}
the squared-variable term is isolated, so we
will take the square root of

each side

_{}
notice the use of the _{} sign, this will give us both a positive and a

negative root

_{}
simplify both sides of the equation, here x
is isolated so we have

solved this equation

Example 2:

_{}
again the squared-variable term is
isolated, so we will take the

square root of each side

_{}
again dont forget the _{} sign, now simplify the radicals

_{}
this time p is not fully
isolated, also notice that _{}4 are

rational numbers, which means

_{} and _{}

_{} and _{}

Example 3:

_{}
squared term is not isolated, add 1
to each side before

beginning

_{}
now take the square root of both sides

_{}
simplify radicals

_{}
radical containing the constant cannot be
simplified, solve for the

variable

_{}
notice the placement of the 1 before the
radical on the

right-hand side, these numbers may not be combined since

1 is a rational number and _{} are irrational numbers

_{}

_{}

_{}

In each of the first 3
examples involving the square root property, notice that there were no
first-degree terms. These equations although they are quadratic in
nature, have the form _{}.
To solve a quadratic equation that contains a first-degree term using the
square root property would involve __ __*
completing the square*
which is another "trick" that will be explained in another lesson.

3)
The third
method for solving quadratic equations described uses the __quadratic formula__.

This formula is _{}. If you notice,
the right hand side has variables a, b, and c.

These variables are the
coefficients of the terms of the quadratic equation. (Remember the
standard form is _{}.)

Example 4:

_{ }
first, the equation must be in standard form

move terms to one side

_{ }
identify a = 1, b = 8, c = 9

_{ }
use the quadratic formula, substitute values

_{}
simplify radical

_{
}
solve for r

_{ } and _{}
quadratic equations have 2 solutions

Example 5:

_{ }
move terms to one side and set equation equal to zero

_{ }
identify a = 5, b = 1, c = 1

_{ }
substitute values

_{ }
simplify radical

_{ }
cannot be simplified further

_{ } and _{}

Remember ... quadratic equations generally have 2
solutions.