Factoring a Trinomial and Completing the Square

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Factoring a Trinomial and Completing the Square

• Lesson 7.3

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Lets review multiplying (x 3)(x 4) we can
use a multiplication rectangle to help us.
x
4
x2
4x
x
12
3x
3
(x 3)(x 4) x2 3x 4x 12 x2 7x 12
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Multiply (x - 2)(x - 5) we can use a
multiplication rectangle to help us.
x
- 5
x2
-5x
x
10
-2x
- 2
(x - 2)(x - 5) x2 - 2x - 5x 10 x2 - 7x 12
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Multiply (3x 2)(2x - 5) we can use a
multiplication rectangle to help us.
2x
- 5
6x2
-15x
3x
- 10
4x
2
(3x 2)(2x - 5) 6x2 4x - 15x - 10 x2 - 11x
- 10
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Try multiplying (4x 1)(3x 5)
Try multiplying (3x - 1)(2x 3)
(4x 1)(3x 5) 12x2 20x 3x 5 12x2
23x 5
(3x - 1)(2x 3) 6x2 9x - 2x - 3 6x2 7x
- 3
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Factoring is the reverse of multiplying. We can
still use the multiplication rectangle to help
us. Suppose we want to factor x2 8x 15
1 x 15 15 1 x 15 15
1 15
-1 -15
3 5
-3 -5
x2
5x
15
3x
You may find it helpful to consider all factors
that make 15.
x2 8x 15 x2 __x ___x 15
(x 5)(x 3)
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Factor x2 2x - 15
1 x (-15) -15 1 x (-15) -15
1 -15
-1 15
3 -5
-3 5
x2
5x
-15
- 3x
You may find it helpful to consider all factors
that make -- 15. Then pick out the pair that
adds to -2
x2 2x - 15 x2 __x ___x - 15
(x 5)(x - 3)
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Factor 3x2 - 2x - 5
3 x (-5) -15 3 x (-5) -15
1 -15
-1 15
3 -5
-3 5
3x2
- 5x
- 5
3x
You may find it helpful to consider all factors
that make -- 15. Then pick out the pair that
adds to -2
3x2 - 2x 5 x2 __x ___x - 15
(3x - 5)(x 1)
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Factor 4x2 - 13x 9
4 x 9 36 4 x 9 36
1 36
-1 -36
2 18
-2 -18
3 12
-3 -12
4 9
-4 -9
6 6
-6 6
4x2
- 4x
- 9x
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Which pair adds to -13?
4x2 - 13x 8 x2 __x ___x 8
(4x - 9)(x - 1)
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Factor these trinomials using the multiplication
rectangle and the sets of factors.
The last three are called perfect squares.
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Complete the Square

• Complete a rectangle diagram to find the product
  (x5)(x5), which can be written (x5)2.
• Write out the four-term polynomial, and then
  combine any like terms you see and express your
  answer as a trinomial.

x2
5x
5x
25
x25x5x25x210x25
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• What binomial expression is being squared, and
  what is the perfect-square trinomial represented
  in the rectangle diagram at right?
• Use a rectangle diagram to show the binomial
  factors for the perfect-square trinomial
  x224x144.

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• Find the perfect-square trinomial equivalent to
  (ab)2 .

• Describe how you can find the first, second, and
  third terms of the perfect square trinomial
  (written in general form) when squaring a
  binomial.

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• Not all polynomials are perfect square
  trinomials, but it is possible to complete the
  square on many of these other polynomials.

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• Consider the expression x26x.
• Where would the x2 fit?
• Where would the 6x fit?
• What would be placed in the last section to have
  a perfect square?
• But we must carefully write this out
  algebraically.

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Lets compare the graphs of
What is the vertex of the parabola?
Which form shows the vertex in the equation?
We call this new form the vertex form for a
parabola. (x-h)2 k
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• Consider the expression x2 8x 4.
• Where would you place x2?
• Where would you place the 8x?
• What number would you like to have in the last
  section to have a perfect square?
• Rewrite the expression x2 8x 4 in the form
  (x-h)2 k.Use a graph or table to verify that
  your expression is equivalent to the original
  expression, x2 8x 4 .

The vertex is located at
(-4,-12)
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• Rewrite each expression in the form (x-h)2k. If
  you use a rectangle diagram, focus on the 2nd-
  and 1st-degree terms first. Verify that your
  expression is equivalent to the original
  expression.

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• When the 2nd-degree term has a coefficient, you
  can first factor it out of the 2nd- and
  1st-degree terms. For example, 3x2 24x5 can be
  written 3(x2 8x)5 . Completing a diagram for x2
  8x can help you rewrite the expression in the
  form a(x-h)2k.

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• Rewrite each expression in the form (x-h)2k.
• Use a graph or table to verify that your
  expression for (a) is equivalent to the original
  expression.

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Example B

• Convert each quadratic function to vertex form.
  Identify the vertex.

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• If you graph the quadratic function yax2bxc,
  what will be the x-coordinate of the vertex in
  terms of a, b, and c?
• How can you use this value and the equation to
  find the y-coordinate?

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Example C

• Nora hits a softball straight up at a speed of
  120 ft/s. If her bat contacts the ball at a
  height of 3 ft above the ground, how high does
  the ball travel? When does the ball reach its
  maximum height?

Using the projectile motion function, you know
that the height of the object at time x is
represented by the equation
. The initial velocity, v0, is 120 ft/s,
and the initial height, s0, is 3 ft. Because the
distance is measured in feet, the approximate
leading coefficient is 16. Thus, the function is
y-16x2 120x 3.
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• To find the maximum height, locate the vertex.
• The softball reaches a maximum height of 228 ft
  at 3.75 s.