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**Unformatted text preview: **Lesson 11
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 ________
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 (
Why not? )? Why or 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 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? 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.
Completing the square:
Given a quadratic equation of the form
that is not
factorable,
1. divide each term by the leading coefficient ( ); skip if leading
coefficient is 1
2. isolate the constant term ( )
3. add the square of half the coefficient of to both sides ( )
4. solve by taking the square root of both sides of the equation,
then isolating the variable
Example 1: Solve the following quadratic equations by completing
the square.
a. b. c. What happens if you take the general form of a quadratic equation
(
) and complete the square?
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 by taking the square root of both sides of the equation,
then isolating the variable
( )
√
√
√ Quadratic Formula:
√ - the result of completing the square for the standard quadratic
equation
- the radicand is called the discriminant
Discriminant:
- the radicand of the quadratic formula
- determines the nature of the solutions (roots)
What does a positive discriminant tell us about our potential
solutions (real, imaginary, how many solutions)? What does a negative discriminant tell us about our potential
solutions (real, imaginary, how many solutions)? What does a discriminant of zero tell us about our potential solutions
(real, imaginary, how many solutions)? Quadratic equations which are not factorable can be solved by either
completing the square or using the quadratic formula; it makes no
difference which method you choose. Example 2: Solve the following by completing the square or by
using the quadratic formula
a. b. c. Example 3: Solve the kinetic energy formula for velocity ( ) Example 4: A manufacturer of ice cream cones wishes to construct
a right circular cone of height 5 inches. Find the radius of the cone
needed to result in a capacity of
. (Round your answer to one
decimal place) Example 5: A rectangular piece of cardboard is 2 inches longer
than it is wide. From each corner, a
inch square is cut out and
the flaps are then folded up to form an open box. If the volume of
the box is
, find the length and width of the original piece of
cardboard. ...

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