
Skills to Master
·
Identify a
quadratic equation and the values of the coefficients corresponding to a, b,
and c in the general form ![]()
· Solve a quadratic equation by factoring
· Use the square root property to solve a quadratic equation for which the value of b is zero
· Solve a quadratic equation by completing the square
· Solve quadratic equations using the quadratic formula
· Know and use the properties of inequality
·
Solve linear inequalities
Quadratic Equations
A quadratic equation is an equation that can be written in the form
, where a, b, and c are real numbers.
Since a quadratic
equation in the above form has two variables that are different terms, we need
a different method of solving than the strategy we employed for linear
equations. You will learn three
different methods to solve quadratic equations.
1.
Solving by
factoring
2.
Solving by
completing the square
3.
Solving by use
of the quadratic formula
Of these three methods,
the first can only be used if the polynomial is factorable. The next two can be used for any quadratics
(assuming that a real number solution exists), but completing the square can
sometimes be more difficult to use than the quadratic formula.
Solving Quadratic Equations by Factoring
If a and b are real numbers, and if ab
= 0, then either a = 0 or b = 0.
Solving by factoring is
a very common method of solving in algebra.
The method makes use of a property of zero:
Remember from a
previous section that factoring a quadratic expression results in the product
of two binomials. That is, a factored
quadratic expression looks something like this:
![]()
where a, b,
c, and d are real numbers. Notice
that this is a multiplication, so if the quadratic equation is equal to zero,
we would have:
. Using the above
property, we can conclude that either
or
. The final step is
finding the value of x in each case
that makes the equation true.
Examples
Solve by factoring: ![]()
Solve by factoring: ![]()
Solve by
factoring: ![]()
Solve by
factoring: ![]()
Solve By Completing the Square
This method is based on
the following types of quadratics:
![]()
![]()
Notice that, regardless
of the value of a, the middle term is
always two times x times a.
Also, the coefficient on the
term must be a 1. This provides a method by which we can produce a perfect square.
Steps for completing
the square:
1.
If the coefficient on
the
term is not a 1, turn it
into a one by dividing through by the coefficient.
2.
Take the value of the
coefficient on the x term, divided it
by two, square this result, and using the additive inverse property, add and
subtract it to the left side of the equation.
3.
Group in parentheses
the
term, the x term, and the positive value from
above. Combine any other like terms as
necessary.
4.
Re-write the grouped
quadratic as the square of the sum of x
and the square root of the positive value from step 2.
Examples
![]()
1.
Notice that step 1 is
not necessary since the coefficient on the
term is already a 1.
2.
The value of the
coefficient on the x term is
_______. Dividing this value by 2 gives
_____. Squaring this value
gives_____. Add zero by changing the
equation to ![]()
3.
Group with parentheses:
![]()
4.
Write as a perfect
square and combine like terms: ![]()
![]()
![]()
Try this one on your
own.
![]()
Solving Quadratic Equations Using the Quadratic Formula
The quadratic formula
can be derived by completing the square on the general form of a quadratic
equation,
. We won’t derive it
here, but it is important to keep two things in mind:
1.
The values the are
input into the formula come from the values of a, b, and c in the given quadratic equation. LEARN TO IDENTIFY THESE VARIABLES!!!
2.
The equation MUST EQUAL
ZERO before the formula can be used!!!
The formula is:
. (You won’t have to
memorize it.)
Examples
Solve using the
quadratic formula: ![]()
Identify the values: a = _____, b = _____, c = _____
The equation is already
equal to zero, we can go straight to the formula:
Solve using the
quadratic formula:
Solve
using the quadratic formula: ![]()
Properties of Inequality
Except
for one very important case, the properties of inequalities are the same as the
properties of equality. This means that,
except for that special case, solving inequalities proceeds in the same way as
solving equalities.
Trichotomy
property For any real numbers a and b, one and only one
of the following must be true:
![]()
Transitive
property For any real numbers a, b, and c:
If
and
, then ![]()
If
and
, then ![]()
Addition/Subtraction
Properties For any real numbers a, b, and c:
If
, then ![]()
If
, then ![]()
The
same properties hold for the greater than relation.
Multiplication/Division
Properties For any real numbers a, b, and c:
If
and
, then
and ![]()
If
and
, then
and ![]()
The
last property is the one to watch out for.
The property tells us that, if an inequality is multiplied or divided by
a NEGATIVE NUMBER, then the inequality sign is reversed. The same properties hold for the greater than
relation.
Examples
Solve
![]()
Solve
![]()
Try
the following on your own.
![]()
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