SOLUTION: Rewrite the quadratic portion of the algebraic expression as the sum or difference of two squares by completing the square. 1/64 − 12x − x^2

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Rewrite the quadratic portion of the algebraic expression as the sum or difference of two squares by completing the square. 1/64 − 12x − x^2       Log On


   

Question 1123174: Rewrite the quadratic portion of the algebraic expression as the sum or difference of two squares by completing the square.

1/64 − 12x − x^2
Found 2 solutions by Edwin McCravy, Theo:
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
The quadratic portion of 

1%2F64+-+12x+-+x%5E2

is just the 2nd and 3rd terms, so
we just work on those and pay no attention
to the 1%2F64 since it is not part of the
quadratic portion:

Write the 2nd and 3rd terms preceded by a - :

1%2F64+-+%28-12x+%2B+x%5E2%29

Swap the terms in the parentheses to get 
the x² term first:

1%2F64+-+%28x%5E2-12x%29

Take half of -12, getting -6,
Then square -6, getting (-6)² or 36, then
Add and subtract 36 inside the parentheses

1%2F64+-+%28x%5E2-12x%2B36-36%29

Factor the first three terms inside the parentheses:

1%2F64+-+%28%28x-6%29%5E2-36%29

Write the 36 as 6²

1%2F64+-+%28%28x-6%29%5E2-6%5E2%29

Since that is the negative of the difference of two
squares, that may not be acceptible as the difference
of two squares.  In that case we distribute the -
to remove the outer parentheses:

1%2F64+-+%28x-6%29%5E2%2B6%5E2%29

The quadratic part is the sum of a negative square
and a positive square, so to make the quadratic
part into the difference of two squares, we reverse
the 2nd and 3rd term so that it will
be the difference of two squares:

1%2F64+%2B+6%5E2+-+%28x-6%29%5E2%29

Now the 2nd and 3rd terms (the quadratic portion)
are the difference of two squares.

Edwin

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
your expression is:

-x^2 - 12x + 1/64

set it equal to y and your equation is:

y = -x^2 - 12x + 1/64

set y equal to 0 and the equation becomes:

0 = -x^2 = 12x + 1/64

add -x^2 - 12x to both sides of the equation to get:

x^2 + 12x = 1/64

take half of the coefficient of the x term and subtract the square of half the coefficint of the x^2 term to form the following equation.

(x+6)^2 - 6^2 = 1/64

add 6^2 to both sides of the equation to get:

(x+6)^2 = 1/64 + 36

simplify to get:

(x+6)^2 = 2305/64

the equation is now in completing the squares form.

take the square root of both sides of this equation to get:

x+6 = sqrt(2305/64)

subtract 6 from both sides to get:

x = sqrt(2305/64) - 6

if that solution is correct, then replacing x with that value in the original equation will make that equation true.

the original equation is -x^2 - 12x + 1/64 = 0

replacing x with sqrt(2305/64) - 6 results in:

-x^2 - 12x + 1/64 = 2.3 * 10^-14.

2.3 * 10^-14 is equal to a decimal point followed by 13 zeros followed by 23.

that looks like .000000000000023 which is a very small number that you can safely assume would be equal to 0 if the calculator had more internal decimal digits to store.

i did confirm manually that the result is accurate and that replacing x with (sqrt(2305/64) - 6) does indeed lead to -x^2 - 12x + 1/64 = 0.

here's a reference on completing the squares method you might find useful.

https://www.purplemath.com/modules/sqrquad.htm