Transcript 6.6 Solving Quadratic Equations

6.6 Solving Quadratic
Equations
Objectives:
1. Multiply binominals using the FOIL
method.
2. Factor Trinomials.
3. Solve quadratic equations by factoring.
4. Solve quadratic equations using the
quadratic formula.
Page 317
• A binomial expression has just two terms (usually an x
term and a constant). There is no equal sign. Its general
form is ax + b, where a and b are real numbers and a ≠
0.
• One way to multiply two binomials is to use the FOIL
method. FOIL stands for the pairs of terms that are
multiplied: First, Outside, Inside, Last.
• This method works best when the two binomials are in
standard form (by descending exponent, ending with the
constant term).
• The resulting expression usually has four terms before it
is simplified. Quite often, the two middle (from the
Outside and Inside) terms can be combined.
For example:
• The opposite of multiplying two binomials is to
factor or break down a polynomial (many termed)
expression.
• Several methods for factoring are given in the text. Be
persistent in factoring! It is normal to try several pairs
of factors, looking for the right ones.
• The more you work with factoring, the easier it will be to
find the correct factors.
• Also, if you check your work by using the FOIL method, it
is virtually impossible to get a factoring problem wrong.
• Remember! When factoring, always take out any
factor that is common to all the terms first.
• A quadratic equation involves a single
variable with exponents no higher than 2.
• Its general form is
where a,
b, and c are real numbers and
.
• For a quadratic equation it is possible
to have two unique solutions, two
repeated solutions (the same number
twice), or no real solutions.
• The solutions may be rational or irrational
numbers.
• To solve a quadratic equation, if it is
factorable:
•
1. Make sure the equation is in the
general form.
•
2. Factor the equation.
•
3. Set each factor to zero.
•
4. Solve each simple linear equation.
To solve a quadratic equation if you can’t factor the
equation:
• Make sure the equation is in the general
form.
• Identify a, b, and c.
• Substitute a, b, and c into the quadratic
formula:
• Simplify.
• The beauty of the quadratic formula is
that it works on any quadratic equation
when put in the form general form.
• If you are having trouble factoring a
problem, the quadratic formula might be
quicker.
• Always be sure and check your solution in
the original quadratic equation.
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Find the product:
Factor
2
x
- 7x + 12.
1. Pairs of numbers which make 12 when
multiplied: (1, 12), (2, 6), and (3, 4).
2. 1 + 12≠7. 2 + 6≠7. 3 + 4 = 7. Thus, d = 3
and e = 4.
3. (x - 3)(x - 4)
4. Check: (x - 3)(x - 4) = x2 -4x - 3x + 12 =
x2 - 7x + 12
2
• Thus, x
- 7x + 12 = (x - 3)(x - 4).
Factor 2x3 +4x2 + 2x.
First, remove common factors: 2x3 +4x2
+2x = 2x(x2 + 2x + 1)
1.
2.
3.
4.
Pairs of numbers which make 1 when multiplied: (1, 1).
1 + 1 = 2. Thus, d = 1 and e = 1.
2x(x + 1)(x + 1) (don't forget the common factor!)
Check: 2x(x + 1)(x + 1) = 2x(x2 +2x + 1) = 2x3 +4x2 + 2x
• Thus, 2x3 +4x2 +2x = 2x(x + 1)(x + 1) =
2x(x + 1)2.
x2 + 2x + 1 is a perfect square trinomial.
The Box Method for Factoring a Polynomial
The Box Method for Factoring a Polynomial
Factor the trinomial:
Use the Quadratic Formula to solve
Solve for x:
Solve for x:
Solve using the quadratic formula:
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Section 6.6 (Read Solving
Quadratic Equation)
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