Unformatted text preview: Lesson 10
Quadratic equation:
 an equation of the form
are real numbers and
 special forms of quadratic equations: , where and The degree of a quadratic equation is ________
Example 1: Determine whether each of the following equations
is quadratic.
a.
b.
c.
d.
)
e. (
f.
g.
In order for an equation to be quadratic, the degree must be
_________. Example 2: If , what must be true about and ? Example 3: If , what must be true about , and ? Zero Factor Theorem:
 the product of two or more factors is zero if and only if at
least one of the factors is zero
if and only if
or
if and only if
,
or
Does this work for any values other than zero (
or Why not? )? Why Solving an equation by factoring:
1. write the equation in polynomial form and set it equal to
zero
2. factor
3. use Zero Factor Theorem
Example 4: Solve the following equations by factoring:
a.
)
b. (
c.
d.
e.
Is there another way to solve parts d. and e.?
If
get just , what could we do to both sides of the equation to
by itself? √ √ Special Quadratic Equation:
 to solve an equation that involves a perfect square, take the
square root of both sides
 this can be used for a single term that is a perfect square, or
a quantity that is a perfect square
 if
, then
√
)
 if (
, then
√ and
√
Do we have to include the sign? Why or why not? Example 3: Solve the following equations by taking the square
root of both sides
a. b. ( c. ( ) ) d. ( ) Why is it that if
? √ , then , but if , then We’ve seen how to solve quadratic equations which are
factorable and quadratic equations which are perfect squares, but
what about quadratic equations which are neither? If a quadratic
equation is not factorable and is not a perfect square, how do we
solve it?
Quadratic equations which are not factorable and are not perfect
squares can be changed into perfect squares by using a
procedure called completing the square.
Examples of Perfect Squares:
(
)
(
)(
(
)
(
)
(
)
(
)
(
) ) When a perfect square is in quadratic form, do you notice a
relationship between the coefficient of the middle term and the
constant term? The constant term is always half the coefficient
of the middle term squared.
( )
( ) ( )
( ( ) ()
( ) ) ()
( )
) ( ( ( )
) Using the pattern from the previous examples, how can we make
a quadratic equation such as
a perfect
square? Completing the square:
Given a quadratic equation of the form
that is
not factorable,
1. divide each term by the leading coefficient ( )
2. isolate the constant term ( )
3. add the square of half the coefficient of to both sides
()
4. solve using the special case (
If the leading coefficient is 1 ( √)
), skip the first step. Example 4: Solve the following equations by completing the
square:
a. b. c. ...
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 Spring '08
 Algebra, Trigonometry, Real Numbers, Equations, Perfect square, Quadratic equations, Quadratic equation, Elementary algebra

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