COMPLETING THE SQUARE
IF WE TRY TO SOLVE this quadratic equation by factoring
x² + 6x + 2 = 0
we cannot. Therefore, we use a technique called completing the square. That means to make the quadratic into a perfect square trinomial, i.e. the form a² + 2ab + b² = (a + b)².
The technique is valid only when 1 is the coefficient of
x².
1) Transpose the constant term to the right:
x² + 6x = −2
2) Add a square number to both sides. Add the square of
half the coefficient of
x. In this case, add the square of 3:
x² + 6x + 9 = −2 + 9
The left-hand side is now the perfect square of (x + 3).
(x + 3)² = 7.
3 is half of the coefficient 6.
This equation has the form
a² | = | b |
which implies |
a | = | ± . |
|
Therefore, |
x + 3 | = | ± |
|
x | = | −3 ± . |
That is, the solutions to
x² + 6x + 2 = 0
are the conjugate pair,
−3 +

, −3 −

.
We can check this. The sum of those roots is −6, which is the negative of the coefficient of x. And the product of the roots is
(−3)² − (

)² = 9 − 7 = 2,
which is the constant term. Thus both conditions on the roots are satisfied. These are the two roots of the quadratic.
x² − 2x | = | 2 |
|
x² − 2x + 1 | = | 2 + 1 |
|
(x − 1)² | = | 3 |
|
x − 1 | = | ± |
|
x | = | 1 ±  |
number b is | b
2 | . Half of 5 is | 5
2 | . Half of | p
q | is | p
2q | . |
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Posted by Muhammad Atif Saeed
on 07:38. Filed under
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Mathandstat
.
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By Muhammad Atif Saeed
on 07:38. Filed under
Gallery
,
Mathandstat
.
Follow any responses to the RSS 2.0. Leave a response