Hi cleanncrazi!!
I start answering this question in the believe that the expression
9/x=3/2 is a result of a typo. If this is not the case, please use the
clarification feature to clarify the relevance of that equation in
this problem and I will rewritte the answer according your
clarification.
1)Choose a problem from your text solve it by completing the square
and then verify your solution using the quadratic formula.
I choose the following exercise:
x^2 + 10x + 21 = 0
The first step is to move the constant term to the right side of the
equation to get:
x^2 + 10x = -21
or
x^2 + 2*5*x = -21
We know that:
x^2 + 2*t*x + t^2 = (x+t)^2
then
x^2 + 2*5*x = x^2 + 2*5*x + 5^2 - 5^2 =
= x^2 + 2*5*x + 25 - 25 =
= (x+5)^2 -25
Then our equation is equivalent to:
(x+5)^2 - 25 = -21 <==> (x+5)^2 = 25-21 = 4
then is:
x+5 = 2 ==> x = -3
or
x+5 = -2 ==> x = -7
The roots of the quadratic equation x^2 + 10x + 21 = 0 are -3 and -7.
Using the Quadratic formula:
Remember that a polynomial a.x^2 + b.x + c , has two possible roots that can be
found using the Quadratic Formula:
x1 = [-b + sqrt(b^2 - 4*a*c)] / (2*a)
and
x2 = [-b - sqrt(b^2 - 4*a*c)] / (2*a)
In this case we have that a = 1, b = 10 and c = 21, then:
x1 = [-10 + sqrt(10^2 - 4*1*21)] / (2*1) =
= [-10 + sqrt(100 - 84)] / (2) =
= [-10 + 4] / 2 =
= -6 / 2 =
= -3
x2 = [-10 - sqrt(10^2 - 4*1*21)] / (2*1) =
= [-10 - sqrt(100 - 84)] / (2) =
= [-10 - 4] / 2 =
= -14 / 2 =
= -7
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2)Write a paragraph comparing and contrasting the two methods
(completing the square and quadratic formula). Explain which method
you prefer and why.
"Completing the square" method is an easy method to solve quadratic
equations and it is useful in all cases. Using the completion of the
square you can obtain the quadratic formula from the general quadratic
equation ax^2 + bx +c .
This means that when you are using the quadratic formula you are using
the completion of the square.
Remember that for the quadratic formula you have to take a square root
to the quantity:
b2 - 4ac
This term is called discriminant and it must be greater than or equal to zero.
Obviously we have three possibilities:
·Discriminant is zero: then both roots are the same (i.e. the
quadratic equation is a perfect square).
·Discriminant is possitive: then we will have two different real roots.
·Discriminant is negative: then we have no roots in the set of the real numbers.
Despite the fact that completing the square is an easy and always
successful method to solve quadratic equations, the quadratic formula'
discriminant (the term (b^2 - 4ac)) gives us inmediate information
about the nature of the roots of the quadratic equation.
Look this example:
x^2 + 2x + 2 = 0
By completing the square we have that:
(x+1)^2 = -1
This is not possible in the set of the real numbers, in other words we
find (after did all the job to complete the square) that the equation
has no root s in the set of the real numbers.
Using the discriminant we find easily and at the first step that:
b^2 - 4ac = 4 - 4*1*2 = 4 - 8 = -4 < 0
Then we know that there are no roots for this equation.
See another example:
x^2 + 2x + 5 = 0
Using the completion of the square:
3x^2 + 2x = -5
x^2 + 2x + 1 = -5 + 1
(x+1)^2 = -4
Then there are no roots for this equation.
Using the quadratic formula:
Discriminant = 2^2 - 4*1*5 = 4-20 = -16 < 0 (no roots)
Or try to complete the square to the following equation:
3x^2 + 7x + 5 = 0 (this equation has no roots, as you can easily see
calculating the discriminant of the quadratic formula)
As we said, the quadratic formula is derived from a completion of the
square, this means that when you use the quadratic formula the
completion of the square has been done and you are using the result of
it, so, in general, you are skipping some steps.
Another thing to keep in account is that the quadratic formula is easy
to reproduce in a spreadsheet or to program in a calculator, because
there is not an algorithm involved like in the completion of the
square.
All these things lead me to choose the quadratic formula as my
preferred way to solve quadratic equations.
For additional reference about this topic see:
"Completing the Square and the Quadratic Formula":
http://www.math.harvard.edu/~engelwar/mathe9/algebra/webAPPD.pdf
"Quadratic Equations - Complete the Square":
http://www.hyper-ad.com/tutoring/math/algebra/Complete_square.html
"Quadratic Formula ? Derivation":
http://www.hyper-ad.com/tutoring/math/algebra/Quadratic_Derivation.html
"COMPLETING THE SQUARE":
http://www.themathpage.com/aPreCalc/complete-the-square.htm
"Completing the Square: Quadratics":
http://www.purplemath.com/modules/sqrquad.htm
"The Quadratic Formula - I":
http://www.purplemath.com/modules/quadform.htm
"The Quadratic Formula - II: The Discriminant: solutions and graphs":
http://www.purplemath.com/modules/quadform2.htm
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3)Find an example from your line of work or daily life that can be
expressed as a quadratic equation. Solve the equation using the method
of your choice.
You may be involved in the following situation:
A friend of you ask for you to loan $20,000 to him and he offer you
two different ways to pay back the money to you. The cost of money is
8%:
At the end of: Year 1 Year 2
Offer 1 : $11,000 $12,000
Offer 2 : $0 $25,000
Which is the best offer:
To know that you must calculate the IRR, to do that you must solve the
following equations:
0 = -20,000 + 11,000/(1+IRR) + 12,000/(1+IRR)^2
0 = -20,000 + 0 + 25,000/(1+IRR)^2
If we call x = 1/(1+IRR) we have:
0 = -20,000 + 11,000x + 12,000x^2
0 = -20,000 + 0 + 25,000x^2
Using the quadratic formula we have for the first equation:
x1 = [-11 + sqrt(121 - 4*12*(-11))]/2*12 = 0.912
x2 = -1.828
Only the possitive value has sense, then:
1/(1+IRR) = 0.912 <==> IRR = 0.0965 = 9.65%
For the second offer we will have that:
x1 = 0.894
x2 = -0.894
Only the possitive value has sense, then:
1/(1+IRR) = 0.894 <==> IRR = 0.1186 = 11.86%
The second offer has greater IRR and also it is greater than the cost
of capital, then this must be the preferred one.
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I hope that this helps you. Feel free to request for any clarification needed.
Best regards.
livioflores-ga |
