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012. `query 12

College Algebra

02-18-2008

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assignment #012

012. `query 12

College Algebra

02-18-2008

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15:10:17

**** query 1.4.28 (was 1.4.18). Explain how you found the real solutions of the equation sqrt(3x+7) + sqrt(x+2) = 1.

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RESPONSE -->

sqrt(3x+7) + sqrt(x+2) =1

[sqrt(3x+7) + sqrt(x+2)]^2= (1)^2

3x+7 + x+2 =1

4x=-8

x=-2

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15:12:11

** Starting with

sqrt(3x+7)+sqrt(x+2)=1

we could just square both sides, recalling that (a+b)^2 = a^2 + 2 a b + b^2. This would be valid but instead we will add -sqrt(x+2) to both sides to get a form with a square root on both sides. This choice is arbitrary; it could be done either way. We get

sqrt(3x+7)= -sqrt(x+2) + 1 . Now we square both sides to get

sqrt(3x+7)^2 =[ -sqrt(x+2) +1]^2. Expanding the right-hand side using (a+b)^2 = a^2 + 2 a b + b^2 with a = -sqrt(x+2) and b = 1:

3x+7= x+2 - 2sqrt(x+2) +1. Note that whatever we do we can't avoid that term -2 sqrt(x+2). Simplifying

3x+7= x+ 3 - 2sqrt(x+2) then adding -(x+3) we have

3x+7-x-3 = -2sqrt(x+2). Squaring both sides we get

(2x+4)^2 = (-2sqrt(x+2))^2.

Note that when you do this step you square away the - sign, which can result in extraneous solutions.

We get

4x^2+16x+16= 4(x+2). Applying the distributive law we have

4x^2+16x+16=4x+8. Adding -4x - 8 to both sides we obtain

4x^2+12x+8=0. Factoring 4 we get

4*((x+1)(x+2)=0 and dividing both sides by 4 we have

(x+1)(x+2)=0 Applying the zero principle we end up with

(x+1)(x+2)=0 so that our potential solution set is

x= {-1, -2}.

Both of these solutions need to be checked in the original equation sqrt(3x+7)+sqrt(x+2)=1

It turns out that the -1 gives us sqrt(4) + sqrt(1) = 1 or 2 + 1 = 1, which isn't true, while -2 gives us sqrr(1) + sqrt(0) = 1 or 1 + 0 = 1, which is true.

x = -1 is the extraneous solution that was introduced in our squaring step.

Thus our only solution is x = -2. **

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RESPONSE -->

&#

Your response did not agree with the given solution in all details, and you should therefore have addressed the discrepancy with a full self-critique, detailing the discrepancy and demonstrating exactly what you do and do not understand about the given solution, and if necessary asking specific questions (to which I will respond).

You particularly need to address the fact that

[sqrt(3x+7) + sqrt(x+2)]^2 is not
3x+7 + x+2

as you asserted.

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15:19:15

**** query 1.4.40 (was 1.4.30). Explain how you found the real solutions of the equation x^(3/4) - 9 x^(1/4) = 0.

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RESPONSE -->

x^(3/4) -9x^(1/4) =0

I don't know how to do this, I don't uderstand the 1/4?

x^(1/4) means the 1/4 power of x, also written as the 4th root of x.
The 3/4 power of x means the 1/4 power, raised to the 3rd power.

Fractional exponents and radicals are covered in Section 1.8.

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15:19:48

** Here we can factor x^(1/4) from both sides:

Starting with

x^(3/4) - 9 x^(1/4) = 0 we factor as indicated to get

x^(1/4) ( x^(1/2) - 9) = 0. Applying the zero principle we get

x^(1/4) = 0 or x^(1/2) - 9 = 0 which gives us

x = 0 or x^(1/2) = 9.

Squaring both sides of x^(1/2) = 9 we get x = 81.

So our solution set is {0, 81). **

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RESPONSE -->

&#

This also requires a self-critique.

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15:26:17

**** query 1.4.46 (was 1.4.36). Explain how you found the real solutions of the equation x^6 - 7 x^3 - 8 =0

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RESPONSE -->

x^6 - 7x^3 -8=0

x^3(x^2 -7x-8)=0

x^3(x+8)(x-1)=0

x+8=0 or x-1=0

x={-8,1}

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15:27:10

** Let a = x^3. Then a^2 = x^6 and the equation x^6 - 7x^3 - 8=0 becomes

a^2 - 7 a - 8 = 0. This factors into

(a-8)(a+1) = 0, with solutions

a = 8, a = -1.

Since a = x^3 the solutions are x^3 = 8 and x^3 = -1.

We solve these equations to get

x = 8^(1/3) = 2 and x = (-1)^(1/3) = -1. **

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RESPONSE -->

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15:31:29

**** query 1.4.64 (was 1.4.54). Explain how you found the real solutions of the equation x^2 - 3 x - sqrt(x^2 - 3x) = 2.

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RESPONSE -->

x^2-3x-sqrt(x^2 - 3x) = 2

x^2-3x = sqrt(x^2 - 3x) +2

(x^2 -3x)^2 = [sqrt(x^2 - 3x) +2]^2

x^4

I don't know

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15:32:10

** Let u = sqrt(x^2 - 3x). Then u^2 = x^2 - 3x, and the equation is

u^2 - u = 2. Rearrange to get

u^2 - u - 2 = 0. Factor to get

(u-2)(u+1) = 0.

Solutions are u = 2, u = -1.

Substituting x^2 - 3x back in for u we get

sqrt(x^2 - 3 x) = 2 and sqrt(x^2 - 3 x) = -1.

The second is impossible since sqrt can't be negative.

The first gives us

sqrt(x^2 - 3x) = 2 so

x^2 - 3x = 4. Rearranging we have

x^2 - 3x - 4 = 0 so that

(x-4)(x+1) = 0 and

x = -4 or x = 1.

DER **

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RESPONSE -->

&#

In your self-critique you need a phrase-by-phrase analysis of the given solution, detailing everything you do and do not understand.
Deconstruct the given solution and explain in detail what you do and do not understand about every part.

. &#

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15:39:01

**** query 1.4.92 \ 90 (was 1.4.66). Explain how you found the real solutions of the equation x^4 + sqrt(2) x^2 - 2 = 0.

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RESPONSE -->

x^4 + sqrt(2) x^2 -2= 0

(x^4 + sqrt(2) x^2 - 2)^2 = (0)^2

x^6 + 2x^4 - 4 = 0

(x^4 + sqrt(2) x^2 - 2)^2 is not
x^6 + 2x^4 - 4 .

To find (x^4 + sqrt(2) x^2 - 2)^2 you need to multiply

(x^4 + sqrt(2) x^2 - 2)(x^4 + sqrt(2) x^2 - 2)

using the distributive law.

Step by step, what do you get when you do this multiplication using the distributive law?

x^3 (x^2 + 2x - 4) = 0

x^3 (x-

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15:40:13

** Starting with

x^4+ sqrt(2)x^2-2=0 we let u=x^2 so that u^2 = x^4:

u^2 + sqrt(2)u-2=0

using quadratic formula

u=(-sqrt2 +- sqrt(2-(-8))/2 so

u=(-sqrt2+-sqrt10)/2

Note that u = (-sqrt(2) - sqrt(10) ) / 2 is negative, and u = ( -sqrt(2) + sqrt(10) ) / 2 is positive.

u = x^2, so u can only be positive. Thus the only solutions are the solutions to

x^2 = ( -sqrt(2) + sqrt(10) ) / 2. The solutions are

x = sqrt( ( -sqrt(2) + sqrt(10) ) / 2 ) and

x = -sqrt( ( -sqrt(2) + sqrt(10) ) / 2 ).

Approximations are x = .935 and x = -.935. **

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RESPONSE -->

I don't understand where the u is coming from. It doesn't make any sense.

Do you understand that

u^2 + sqrt(2)u-2=0

is in quadratic form?

Do you understand that if we let u=x^2 the equation

u^2 + sqrt(2)u-2=0

becomes

x^4+ sqrt(2)x^2-2=0?

Which phrases, which lines, which arguments in the given solution do you and do you not understand?

You do not appear to be self-critiquing your work. See my notes. You need to self-critique so I can tell what you do and do not understand. Then I can insert comments or questions to help you understand those things you need help with.

I have posed a few questions related to some of your work.

&#Try deconstructing the given solution to identify what you do and do not understand. Just send a copy of the question, your answer, the given solution and the self-critique. &#

Assign 12

&#

You particularly need to address the fact that [sqrt(3x+7) + sqrt(x+2)]^2 is not 3x+7 + x+2 as you asserted.

x^(1/4) means the 1/4 power of x, also written as the 4th root of x. The 3/4 power of x means the 1/4 power, raised to the 3rd power. Fractional exponents and radicals are covered in Section 1.8.

&#

This also requires a self-critique.

&#

In your self-critique you need a phrase-by-phrase analysis of the given solution, detailing everything you do and do not understand. Deconstruct the given solution and explain in detail what you do and do not understand about every part.

(x^4 + sqrt(2) x^2 - 2)^2 is not x^6 + 2x^4 - 4 . To find (x^4 + sqrt(2) x^2 - 2)^2 you need to multiply (x^4 + sqrt(2) x^2 - 2)(x^4 + sqrt(2) x^2 - 2) using the distributive law. Step by step, what do you get when you do this multiplication using the distributive law?

Do you understand that u^2 + sqrt(2)u-2=0 is in quadratic form? Do you understand that if we let u=x^2 the equation u^2 + sqrt(2)u-2=0 becomes x^4+ sqrt(2)x^2-2=0? Which phrases, which lines, which arguments in the given solution do you and do you not understand?

You do not appear to be self-critiquing your work. See my notes. You need to self-critique so I can tell what you do and do not understand. Then I can insert comments or questions to help you understand those things you need help with. I have posed a few questions related to some of your work.

&#Try deconstructing the given solution to identify what you do and do not understand. Just send a copy of the question, your answer, the given solution and the self-critique. &#

You particularly need to address the fact that

[sqrt(3x+7) + sqrt(x+2)]^2 is not
3x+7 + x+2

as you asserted.

x^(1/4) means the 1/4 power of x, also written as the 4th root of x.
The 3/4 power of x means the 1/4 power, raised to the 3rd power.

Fractional exponents and radicals are covered in Section 1.8.

In your self-critique you need a phrase-by-phrase analysis of the given solution, detailing everything you do and do not understand.
Deconstruct the given solution and explain in detail what you do and do not understand about every part.

(x^4 + sqrt(2) x^2 - 2)^2 is not
x^6 + 2x^4 - 4 .

To find (x^4 + sqrt(2) x^2 - 2)^2 you need to multiply

(x^4 + sqrt(2) x^2 - 2)(x^4 + sqrt(2) x^2 - 2)

using the distributive law.

Step by step, what do you get when you do this multiplication using the distributive law?

Do you understand that

u^2 + sqrt(2)u-2=0

is in quadratic form?

Do you understand that if we let u=x^2 the equation

u^2 + sqrt(2)u-2=0

becomes

x^4+ sqrt(2)x^2-2=0?

Which phrases, which lines, which arguments in the given solution do you and do you not understand?

You do not appear to be self-critiquing your work. See my notes. You need to self-critique so I can tell what you do and do not understand. Then I can insert comments or questions to help you understand those things you need help with.

I have posed a few questions related to some of your work.