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Perfect square trinomials:  Level 2

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(a + b

The square of a trinomial

Completing the square

Problem 8.   Without multiplying out

a)  explain why (1 − x)² = (x − 1)².

Because (1 − x) is the negative of (x − 1). And (−a)² = a² for any quantity a.

b)  explain why (1 − x)³ = −(x − 1)³.

(−a)³ = −a³ for any quantity a.

*

The following problems show how we can go from what we know to what we do not know.

Problem 9.    Use your knowledge of  (a + b)²  to multiply out (a + b)3.

[Hint:   (a + b)3 = (a + b)(a + b)²]

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  (a + b)(a + b = (a + b)(a² + 2ab+ b²)
 
  = a3 + 2a²b + ab²
 
    + a²b  + 2ab² + b3
 
(a + b)3 = a3 + 3a²b  + 3ab² + b3

Problem 10.   Multiply out  (x + 2)3.

  (x + 2)(x + 2)² = (x + 2)(x² + 4x+ 4)
 
  = x3 + 4x² + 4x
 
    + 2x² + 8x + 8
 
(x + 2)3 = x3 + 6x² + 12x + 8

Problem 11.   Multiply out  (x − 1)3.

  (x − 1)(x − 1)² = (x− 1)(x² − 2x + 1)
 
  = x3 − 2x² + x
 
    x² + 2x − 1
 
(x − 1)3 = x3 − 3x² + 3x − 1

Problem 12.  The square of a trinomial.   Use your knowledge of
(a + b)² to multiply out  (a + b + c)².

[Hint:  Treat   as a binomial with as the first term.]

Show that it will equal the sum of the squares of each term, plus twice the product of all combinations of the terms.

( + c = (a + b)² + 2(a + b)c + c²
 
  = a² + 2ab + b²  +  2ac + 2bc  + c²
 
= a² + b² + c² + 2ab + 2ac + 2bc

Problem 13.   Can you generalize the result of the previous problem?   Can you immediately write down the square of  (a + b + c + d)?

(a + b + c + d)² =  a²  + b² + c² + d²
 
    + 2ab + 2ac + 2ad + 2bc + 2bd + 2cd

Completing the square

x² + 8x + _?_ = (x + _?_

When the coefficient of x² is 1, as in this case, then to complete a perfect square trinomial, we must add a square number.  What square number must we add?

We must add the square of half of coefficient of x.  The trinomial will then be the square  of x plus half that coefficient.

x² + 8x + 16 = (x + 4

We add the square of half the coefficient of x -- which in this case is 4 -- because when we multiply (x + 4)², the coefficient of x will be twice that number.


Problem 14.   

  a)  How do we indicate half of any number b?   b
2
  b)  How do we indicate half of any fraction  p
q
?    p 
2q

(Skill in Arithmetic, Lesson 26.)

Example 7.   Complete the square:  x² − 7x + ? = (x − ?)²

  Solution.  We will add the square of half of 7, which we write as  7
2
.
  x² − 7x 49
 4
  =   (x −  7
2

And since the middle term of the trinomial has a minus sign, then the binomial also must have a minus sign.

Problem 15.   Complete the square.  The trinomial is the square of what binomial?

a)  x² + 4x + ?   x² + 4x + 4 = (x + 2)²

b)  x² − 2x + ?   x² − 2x + 1 = (x − 1)²

c)  x² + 6x + ?   x² + 6x + 9 = (x + 3)²

d)  x² − 10x + ?   x² − 10x + 25 = (x − 5)²

e)  x² + 20x + ?   x² + 20x + 100 = (x + 10)²

  f)  x² + 5x + ?   x² + 5x 25
 4
 = (x 5
2
  g)  x² − 9x + ?   x² − 9x 81
 4
 = (x −  9
2
  h)  x² + bx + ?   x² + bx b²
 = (x b
2
  i)  x² +  b
a
x + ?     x² +  b
a
x  b²
4a²
 = (x  b 
2a

Back to Section 1


Next Lesson:  The difference of two squares


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