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Note that in all cases of the the perfect square, square,
Completing the square
if ( a = 1),
()
c=
then
One general form of the quadratic expression is: 2
ax
+bx + c
b
2
2
the end term is the square of one-half the coefficient of the middle term.
Perfect Squares: ( case a = 1) Consider the Expression below where: ( a = 1, b = 6 and c = 9)
Incomplete squares 3
x x
2
+ 6x + 9 9
End ( last) Term
First Term
Middle Term
The incomplete square can be viewed as a complete square with a portion removed. If we factorize this expression we get : x
2
+
6x
+
9
Example1: What must be added to square:
2
( x + 3)
=
(x + 3)
x
3
3x
9
x2
3x
2
+ 8x to make it a perfect
x
which is called a complete complete ( or perfect ) square square because it can be visualized as a square with each side = (x + 3)
(x + 3)
x
4
3
(x + 3)2 x
Solution:
a = 1, b = 8 , c = ?
where
( x +3)( x +3)
= =
2
+3x +3x +3 x +6x + 9
x
2
For a perfect square: if a = 1, then
2
so we must add c = Other examples examples of complete (perfect) (perfect) squares are:
•
x
•
x
•
x
2
2
=
( x + 2)
+10x +25 =
( x + 5)2
+4x + 4
2
2
+12x +36 =
()
( x +6)2
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Giving
x
2
+8x + 16 =
8 2
()
c=
b
2
2
2
=
4
2
= 16
( x + 4)2 .. perfect square
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2
Incomplete Squares contd
Functions of the form :
In example1 , x 2+ 8x can be considered an incomplete square, where we showed that by adding 16, we could complete the square to create a perfect square). That is:
Any general Quadratic expression in the form ax 2+ bx+ c 2 can be converted to a new form a ( x + h ) + k by first expressing h and k in terms of a, b and c.
2
+ 8x x ⏟
+16
= ( x + 4)
2
ax
+ bx+ c
The formulas for h and k are determined as shown below.
eqn 2
Finding h and k
Now if we re-arrange eqn 2, we can see that: x
2
2 +8x = ( x + 4) - 16 2
= ( x + 4) − 2
= ( x +h) −
4
2
ax
... eqn 1
2
h
2
b
( h= )
where
2
=
8 2
Step
This is an important conclusion which may be generalized as follows:
h=
+ bx = ( x +h) 2−h2 where
Result
Factor a from the first
2
a
x
two terms of eqn.1
+
b a
+c
x
Complete the square of
2 2
Action
= 4
1
x
bx c
2
x
b 2
3
2
the terms in bracket
+
b a
( x + b ) −(
2
b
2a
2a
)
x
Place the result of step 2 back into step 1
( x +
a
2
b
2a
) −(
2
b
2a
) +c
This is known as completing the square; *****************************************************
x
2
+12x
( x +h)2− h2
in the form
4
Solution: h
b = 12 , so
⇒
x
=
b 2
=6
2 2 + 12x = ( x +6) −6
4a
2
2
b
=
a ( x +
=
a ( x +
=
a ( x + h )
Simplify
2
) − ab + c
2a
Example2:
Express
2
b
a ( x +
2
) + c− ab2
2a
4a
2
) + 4ac− b
b
2a
2
4a
2
2
+ k
2
= ( x + 6) − 36
h=
*****************************************************
Example3 2
Express now:
5 x – 7x
h=
7 2
in the form
2
( x + h) −h
⇒
x – 7x
2a
Compare k=
2
4ac – b
2
4a
= 3.5 Giving: h =
2
b
2
2
=
( x −3.5) −3.5
=
( x −3.5)2 −12.25
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b 2a
and
k=
4ac – b 4a
2
nb: Students should memorize these two formulas 2
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Example 4
Express
x
Example 7 2
+7x −13 in the form ( x + h) 2+k
−2x 2− 7x+ 5
Express Solution:
h=
b 2a
k =
and
7
=
4ac – b 4a
= 3.5
2
2
h=
4× 1 ×(−13 )−7 4 ×1
=
2
=
x
2
+7x −13 =
2
k −a ( x + h )
( a = -2, b = - 7, c = 5)
b
2a
=
−7 = −4
1.75
− 25.5 and
so;
in the form
k =
4ac – b
2
4a
4 ×−2 × 5
=
−(−7)
2
= 11.125
4×−2
( x +3.5)2 − 25.5 − 2x2− 7x+ 5
so:
=
2
11.125 − 2 ( x + 1.75)
Example 5:
Express the function 1− 5x − x 2
2
k −( x + h )
in the form
Solution:
b
2a
Express in the form: 2
a ( x + h )
Using the Formulas: ( a = -1, b = -5, c = 1) h=
Practice questions:
=
−5 = 2.5 −2
4ac – b
and
k =
so
1− 5x − x
2
=
4a 2
=
4 ×−1 ×1 −(−5 )
Express in the form: 2
+ k
k − a ( x + h )
2
6.
− x 2− 2x+ 3
−7x −1
7.
− 2x2+ x + 6
+ 2x − 7
8.
3+ 2x − 4x
2
9.
1+ x − x
− 6x + 1
1.
2x
2.
x
3.
3x
4.
1 − 5x + 2x
5.
2 − 3x + x
2
2
2
2
2
4×−1
= 7.25
2
10.
2− x −2x
2
2
7.25 − ( x + 2.5)
Answers :Exercise A1
Functions of the form
ax
2
( a ≠1 )
+ bx+ c
Example 6:
Express
2x
2
−6x −13 in the form
2
a ( x + h )
+ k
Solution ( a = 2, b = - 6, c = -13) h=
and
b
2a
=
4ac – b k = 4a 2x
2
−6 4 2
=
−1.5
4 × 2×(− 13 ) −(− 6) = 4× 2
2
#
a
h
k
1
2
-1.5
-3.5
2
1
-3.5
-13.25
3
3
0.33
-7.33
4
2
-1.25
-2.125
5
1
-1.5
-0.25
6
-1
1
4
7
-2
-0.25
6.125
8
-4
-0.25
3.25
9
-1
-0.5
1.25
10
-2
0.25
2.125
= −17.5
−6x −13 = 2 ( x − 1.5 )2 − 17.5 Topic discussions in the math club (forum)
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