Transcript Factoring quadratics 1

Warm-Up Exercises
Find the product.
1. (x + 6)(x – 4)
ANSWER
x2 + 2x – 24
2. (2y + 3)( y + 5)
ANSWER
2y2 + 13y + 15
Warm-Up Exercises
Find the product.
3. The dimensions of a rectangular print can be
represented by x – 2 and 2x + 1. Write an expression
that models the area of the print. What is its area if
x is 4 inches?
ANSWER
2x2 – 3x – 2; 18in.2
EXAMPLE
Warm-Up1Exercises
Factor when b and c are positive
Factor x2 + 11x + 18.
SOLUTION
Find two positive factors of 18 whose sum is 11.
Make an organized list.
Factors of 18
Sum of factors
18, 1
18 + 1 = 19
9, 2
9 + 2 = 11
6, 3
6+3=9
Correct sum
EXAMPLE
Warm-Up1Exercises
Factor when b and c are positive
The factors 9 and 2 have a sum of 11, so they are the
correct values of p and q.
ANSWER
x2 + 11x + 18 = (x + 9)(x + 2)
CHECK
(x + 9)(x + 2) = x2 + 2x + 9x + 18 Multiply binomials.
= x2 + 11x + 18
Simplify.
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 1
Factor the trinomial
1.
x2 + 3x + 2
ANSWER (x + 2)(x + 1)
2.
a2 + 7a + 10
ANSWER (a + 5)(a + 2)
3.
t2 + 9t + 14.
ANSWER (t + 7)(t + 2)
EXAMPLE
Warm-Up2Exercises
Factor when b is negative and c is positive
Factor n2 – 6n + 8.
Because b is negative and c is positive, p and q must
both be negative.
Factors of 8
Sum of factors
–8, –1
–8 + (–1) = –9
–4, –2
–4 + (–2) = –6
ANSWER
n2 – 6n + 8 = (n – 4)( n – 2)
Correct sum
EXAMPLE
Warm-Up3Exercises
Factor when b is positive and c is negative
Factor y2 + 2y – 15.
Because c is negative, p and q must have different
signs.
Factors of –15
–15, 1
15, –1
–5, 3
5, –3
ANSWER
Sum of factors
–15 + 1 = –14
15 + (–1) = 14
–5 + 3 = –2
5 + (–3) = 2
y2 + 2y – 15 = (y + 5)( y – 3)
Correct sum
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 2 and 3
Factor the trinomial
4. x2 – 4x + 3.
ANSWER
(x – 3)( x – 1)
5. t2 – 8t + 12.
ANSWER
(t – 6)( t – 2)
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 2 and 3
Factor the trinomial
6. m2 + m – 20.
ANSWER
(m + 5)( m – 4)
7. w2 + 6w – 16.
ANSWER
(w + 8)( w – 2)
Warm-Up4Exercises
EXAMPLE
Solve a polynomial equation
Solve the equation x2 + 3x = 18.
x2 + 3x = 18
x2 + 3x – 18 = 0
(x + 6)(x – 3) = 0
x +6=0
or x – 3 = 0
x = – 6 or
x=3
Write original equation.
Subtract 18 from each side.
Factor left side.
Zero-product property
Solve for x.
ANSWER The solutions of the equation are – 6 and 3.
Warm-Up
EXAMPLE
4Exercises
for Example
4
Solve a polynomial
equation
GUIDED
PRACTICE
8.
Solve the equation s2 – 2s = 24.
ANSWER The solutions of the equation are – 4 and 6.
Warm-Up5Exercises
EXAMPLE
Solve a multi-step problem
BANNER DIMENSIONS
You are making banners to hang
during school spirit week. Each
banner requires 16.5 square feet
of felt and will be cut as shown.
Find the width of one banner.
SOLUTION
STEP 1
Draw a diagram of two banners together.
Warm-Up5Exercises
EXAMPLE
Solve a polynomial equation
STEP 2
Write an equation using the fact that the area of 2 banners
is 2(16.5) = 33 square feet. Solve the equation for w.
A=l w
Formula for area of a rectangle
33 = (4 + w + 4) w
Substitute 33 for A and (4 + w + 4) for l.
0 = w2 + 8w – 33
Simplify and subtract 33 from each side.
0 = (w + 11)(w – 3) Factor right side.
w + 11 = 0 or w – 3 = 0 Zero-product property
w = – 11 or w = 3 Solve for w.
The banner cannot have a negative width,
ANSWER
so the width is 3 feet.
Warm-Up
Exercises
GUIDED
PRACTICE
9.
for Example 5
WHAT IF? In example 5, suppose the area of a banner
is to be 10 square feet. What is the width of one
banner?
ANSWER
2 feet
Daily
Homework
Quiz
Warm-Up
Exercises
Factor the trinomial.
1.
x2 – 6x – 16
ANSWER
2.
y2 + 11y + 24
ANSWER
3.
(x +2)(x – 8)
(y +3)(y + 8)
x2 + x – 12
ANSWER
(x +4)(x – 3)
Daily
Homework
Quiz
Warm-Up
Exercises
4. Solve a2 – a = 20
ANSWER
– 4, 5
5. Each wooden slat on a set of blinds has width w
and length w + 17. The area of one slat is 38 square
inches. What are the dimensions of a slat?
ANSWER
2 in. by 19 in.