3x^2(4x-12)^2 + x^3(2)(4x-12)(4) HALP

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In summary, the conversation discusses factoring an expression using the "product rule" in calculus. The expert summarizer provides a step-by-step process for factoring and simplifying the expression, and also mentions the concept of "difference of two squares" that can be applied to similar problems.
  • #1
Hierophant
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3x^2(4x-12)^2 + x^3(2)(4x-12)(4)
Factor this expression completely. This type of question occurs in calculus in using the "product rule".

Attempt:
I first factor out a x^2 (4x-12) factor out the 4 from this, then have 4x^2 (X-3)
The left overs are: 3(4x-12) + x(2)(4)
I factor out a 4 from the left overs, I now have. 16x^2(x-3) 3(x-3) + x(2)
Then I simplify: 16x^2 (x-3) 3(x-3) +2x = 16x^2 (x-3) (5x-9)


Which is the right answer, but I may have been biased into finding this as I did know the answer before hand. So is my reasoning sound? Maybe you could go through this yourself and show me your process, practice does not hurt :)
 
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  • #2
When you factor out x^2 (4x-12) you have:
[x^2 (4x-12)]*[3(4x-12) + 8x] ... which is what you got. Then you factor out the 4:
[16 x^2 (x-13)]*[3(x-3)+2x] => [16 x^2 (x-3)]*[3x - 9 +2x] => [16 x^2 (x-3)]*[5x-9]
And then discard the extra brackets.

Like you, I did not notice the extra factor of 4 in (4x-12) on the first pass.
 
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  • #3
Thanks for the reply. I have another question...

Factor this completely.

4a^2c^2 - (a^2 - b^2 + c^2)^2

I am so stumped on this one.

Thanks again for the previous answer.
 
  • #4
Square the second term ... you should get six terms. One of these six can be combined with the first term of the original expression.

Then look for simplifications ...
 
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  • #5
Hierophant said:
Thanks for the reply. I have another question...

Factor this completely.

4a^2c^2 - (a^2 - b^2 + c^2)^2

I am so stumped on this one.

Thanks again for the previous answer.

You often get things like this in exercises and even sometimes in real science problems. This is a "difference of two squares". You've probably seen the factorising of that. After using that there might be some further simplification possible.
 
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1. What is the purpose of the equation "3x^2(4x-12)^2 + x^3(2)(4x-12)(4) HALP"?

The purpose of this equation is to represent a mathematical expression that can be solved to find the value of x. It is a combination of different operations such as multiplication, exponentiation, and addition.

2. What does the notation "x^2" mean in the equation?

The notation "x^2" means x raised to the power of 2, or x multiplied by itself twice. In other words, it is the same as x*x or x^2.

3. How can "4x-12" be raised to the power of 2?

In this equation, "4x-12" is being multiplied by itself, which is the same as raising it to the power of 2. This can also be represented as (4x-12)(4x-12) or (4x-12)^2.

4. What is the purpose of the term "HALP" in the equation?

The term "HALP" is most likely used as a mnemonic or a reminder to the person solving the equation to ask for help if they are having trouble solving it. It does not have a mathematical significance in the equation itself.

5. Can this equation be simplified further?

Yes, this equation can be simplified by using the distributive property to expand the terms and combining like terms. The final simplified form will depend on the value of x and the specific instructions given for solving the equation.

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