Tutor: Hi! I am now working on your question and you will soon see me writing on your drawing board.
Tutor: Ok... Let me write down the problem...
Tutor: :)
Student: Ok i have a few to work on
Tutor: Did I write the question correctly?
Student: Yes
Tutor: Ok...so first we'll change the division sign into multiplication
Student: The question states to simplify the complex question
Student: Ok
Tutor: Now a^2 - b^2 = ( a + b ) ( a - b )
Tutor: (a + b) gets cancelled...
Tutor: Shall i move to the next sheet?
Student: Yes
Tutor: Now dividing each term of the numerator by the denominator separately...we get...
Student: I dont understand how you got a^4
Tutor: See we have in the numerator (a-b)(a^2+b^3)
Tutor: Sorry
Tutor: It should be a^3
Student: And how did you get ab^3
Tutor: When we multiply (a-b)(a^2 + b^3)
Tutor: We get
Student: Ok
Tutor: A x a^2 + a x b^3 - a^2 x b - b x b^3
Tutor: Got it?
Tutor: :)
Student: Yes thanks
Student: Do i ask you my next question on different problems or go to refresh?
Tutor: You may ask me your problem...no need to refresh...
Tutor: Thi would be the solution..
Tutor: Are there any more questions that I can help you with?
Student: Ok next problem is simplify the complex fraction:
Student: 5/2y + 3/4 divided by 3/4 -2/y
Student: The Y is on the bottom at 2y
Tutor: Ok...
Student: Yes
Tutor: Ok...so first we'll simplify both the brackets...
Tutor: We have made the denominator common for both the brackets...
Tutor: And we'll change the division sign into multiplication by taking reciprocal of the right term
Tutor: 4Y gets cancelled...
Tutor: This will be the answer... :)
Tutor: Any questions regarding the solution?
Student: Yes, why wouldn't 3y get cancelled? because they are both +?
Tutor: The 3Y is not alone... the numerator is 10 + 3Y
Tutor: If it would have been as a product like 10 x 3Y
Tutor: Then would been possible to cancel it...
Tutor: Similarly in the denominator...
Student: Ok
Tutor: The #Y term is present WITH the term -8
Tutor: We can only cancel terms which exist independently in the numerator or denominator...
Student: Got it. are you ready for more
Tutor: Yes...
Tutor: :)
Student: To factor completely 81x^2-16y^2
Tutor: Here, if we notice we will realise that 81 and 16 are perfect squares...
Student: Right
Tutor: Can you tell me 81 is the perfect square of which number?
Student: 9
Tutor: Correct!!
Tutor: And 16?
Student: 16 is perfect sq 0f 4
Tutor: Great!!
Tutor: So we can rewrite the given expression as...
Student: That is the answer right?
Tutor: Now we have an expression which is of the form :
A^2 - B^2 = (A + B)(A - B)
Tutor: No...it can be factoprized further...
Tutor: This is the final answer... :)
Student: Got it!
Tutor: Great!! :)
Tutor: Are there any more questions that I can help you with?
Student: Next is factor completely: -3x^5-18x^4-24x^3
Student: Yes
Tutor: Now if we notice, all the three terms have (-3X^3) common...
Tutor: So we'll take it out common outside the brackets...
Tutor: Any questions how we got this?
Student: No i understand i just always want to make 2 bracket instead of one
Student: That is the furthest we can factor right?
Tutor: Like this?
Tutor: Now we can further solve this expression...
Tutor: Here we have split up 6X as 4X + 2X
Tutor: This is the answer... :)
Tutor: Any questions regarding the solution?
Student: Ok, i think i just miss a step and get confused
Tutor: Which step did you miss?
Student: Next is 2x^3-x^2-8x+4=0 solve for x
Student: Factoring the 6x
Tutor: Ok :)
Student: You got it
Tutor: Taking X^2 common out of the first 2 terms and (-4) out of the last 2 terms, we get...
Tutor: Now for this product to be equal to zero, either of the brackets is equal to zero....
Tutor: Any questions so far?
Student: I would have thought x^2 . 2x =2x^3
Tutor: Yeah that is true...
Tutor: That is why when we take X^2 common, 2X remains inside the bracket...
Tutor: These are the values of X
Tutor: X has three values... +2,-2, 1/2
Tutor: Are there any more questions that I can help you with?
Student: So x= 2,-2,1/2?
Tutor: Correct!
Tutor: Yes
Student: Now i just want to check my answers: write in lowest terms:
Student: 12x^2-24x divided by 6x^2-6x-12
Tutor: Shall i solve it for you or you only want the answer?
Student: If its quicker for you to just solve that fine
Tutor: I'll solve it for you :)
Tutor: Taking 12 X common in the numerator and 6 common in the denominator...
Student: I got 2x^2-4x-12
Tutor: Close...but the answer is something else...let me show you...
Tutor: This is the answer...
Tutor: :)
Tutor: Any questions regarding the solution?
Student: No
Student: How about divide & simplify:
Tutor: Are there any more questions that I can help you with?
Tutor: Ok...
Student: X^2+5x over x^2 divided by x^2 over x-5
Student: Sorry on the first one it should be x^2 -25
Tutor: Is it correct now?
Student: No it should be x^2+5x over x^2-25 and the rest is right
Tutor: Ok...
Student: Got it
Tutor: This is the answer... :)
Tutor: Are there any more questions that I can help you with?
Student: This next one seem easy but i cant understand it 5 over p-3 subtracted by 2 over 3-p and it says to add
Tutor: Is this the question?
Student: Yes
Tutor: Ok..
Tutor: Notice that if we reverse the denominator of the second term, we'll have same denominator in both the terms...
Tutor: To reverse the denominator, we need to make it negative...
Tutor: Any questions so far?
Student: Nope
Tutor: This is the answer :)
Student: Great thats easy
Tutor: Are there any more questions that I can help you with?
Student: Now to subtract: 2x over x^2-1 subtracted by 5 over x^2-x-2
Tutor: Is this right?
Student: Yes
Tutor: Here we have factorized both the denominators...
Tutor: Any questions so far?
Student: No
Tutor: This is the answer... :)
Tutor: Are there any more questions that I can help you with?
Student: So the x+1 and the x-1 cant cancel because they are both denominators right?
Tutor: Yeah they can't cancel...
Student: How about factor completely: 6x^3 + 5x^2 -4x
Student: Are you tired yet?
Tutor: I'm not tired!! :)
Tutor: Taking X common...
Tutor: Now writing 5X = 8X - 3X
Tutor: This will be the answer... :)
Tutor: Are there any more questions that I can help you with?
Student: How did you get 8x?
Tutor: 5X = 8X - 3X
Tutor: See the quadratic awas 6X^2 + 5X - 4
Student: Yes
Tutor: We first find the product of the coefficient of X^2 and the constant
Tutor: The coefficient is 6
Tutor: And the constant term is (-4)
Tutor: So the product is (-24)
Tutor: Now we need to find two such factors of 24 whose difference is 5
Student: Ok
Student: Ok next?
Tutor: Now 24 = 1 x 24
2 x 12
3 x 8
4 x 6
Tutor: Out of these...which are the two factors whose difference is 5?
Student: 4x6?
Tutor: No... the difference between the 2 factors... like in 4 x 6 ...the difference is 2
Tutor: 6-4 =2
Tutor: The factors whose difference is 5 are 3 x 8
Student: Ok so it would be 8x3
Tutor: Because 8-3 =5
Tutor: Therefore we have splitted 5X = 8X - 3X
Student: Ok i get it now
Tutor: For example take the previous question...
Tutor: In which we had to factorize X^2 - X -2
Tutor: Here the product of the coefficient of X^2 is 1 and the constant is (-2)
Tutor: So the product is -2
Tutor: 1 x (-2) = -2
Tutor: Now we have to find two such factors of 2 whose difference is 1
Tutor: Since the coefficient of X is 1
Tutor: now 2 = 1 x 2
Tutor: That is the only possible combination...
Tutor: And this is the one which we require since 2 - 1 = 1
Tutor: Therefore we split -X = -2X + X
Tutor: Now do you get it? how we split the middle term ?
Student: Yes
Tutor: Great!! :)
Tutor: Are there any more questions that I can help you with?
Student: Yes the next question is 7x-4 over 5x + 4 over x = 9over 5 solve for x
Tutor: Is this right?
Student: No, 7x -4 over 5x then + 4/x = 9/5
Tutor: OK...
Student: The 4 goes on the bottom over the x
Student: Yes
Tutor: First we'll make the denominator common...
Tutor: Removing the denominator from the whole equation, we get...
Tutor: This is the answer :)
Tutor: Are there any more questions that I can help you with?
Student: That should do it, thanks for all of your help!!
Tutor: You are always welcome :)
Tutor: Good bye. Thank you for using Instant Math Help.
Tutor: Ok... Let me write down the problem...
Tutor: :)
Student: Ok i have a few to work on
Tutor: Did I write the question correctly?
Student: Yes
Tutor: Ok...so first we'll change the division sign into multiplication
Student: The question states to simplify the complex question
Student: Ok
Tutor: Now a^2 - b^2 = ( a + b ) ( a - b )
Tutor: (a + b) gets cancelled...
Tutor: Shall i move to the next sheet?
Student: Yes
Tutor: Now dividing each term of the numerator by the denominator separately...we get...
Student: I dont understand how you got a^4
Tutor: See we have in the numerator (a-b)(a^2+b^3)
Tutor: Sorry
Tutor: It should be a^3
Student: And how did you get ab^3
Tutor: When we multiply (a-b)(a^2 + b^3)
Tutor: We get
Student: Ok
Tutor: A x a^2 + a x b^3 - a^2 x b - b x b^3
Tutor: Got it?
Tutor: :)
Student: Yes thanks
Student: Do i ask you my next question on different problems or go to refresh?
Tutor: You may ask me your problem...no need to refresh...
Tutor: Thi would be the solution..
Tutor: Are there any more questions that I can help you with?
Student: Ok next problem is simplify the complex fraction:
Student: 5/2y + 3/4 divided by 3/4 -2/y
Student: The Y is on the bottom at 2y
Tutor: Ok...
Student: Yes
Tutor: Ok...so first we'll simplify both the brackets...
Tutor: We have made the denominator common for both the brackets...
Tutor: And we'll change the division sign into multiplication by taking reciprocal of the right term
Tutor: 4Y gets cancelled...
Tutor: This will be the answer... :)
Tutor: Any questions regarding the solution?
Student: Yes, why wouldn't 3y get cancelled? because they are both +?
Tutor: The 3Y is not alone... the numerator is 10 + 3Y
Tutor: If it would have been as a product like 10 x 3Y
Tutor: Then would been possible to cancel it...
Tutor: Similarly in the denominator...
Student: Ok
Tutor: The #Y term is present WITH the term -8
Tutor: We can only cancel terms which exist independently in the numerator or denominator...
Student: Got it. are you ready for more
Tutor: Yes...
Tutor: :)
Student: To factor completely 81x^2-16y^2
Tutor: Here, if we notice we will realise that 81 and 16 are perfect squares...
Student: Right
Tutor: Can you tell me 81 is the perfect square of which number?
Student: 9
Tutor: Correct!!
Tutor: And 16?
Student: 16 is perfect sq 0f 4
Tutor: Great!!
Tutor: So we can rewrite the given expression as...
Student: That is the answer right?
Tutor: Now we have an expression which is of the form :
A^2 - B^2 = (A + B)(A - B)
Tutor: No...it can be factoprized further...
Tutor: This is the final answer... :)
Student: Got it!
Tutor: Great!! :)
Tutor: Are there any more questions that I can help you with?
Student: Next is factor completely: -3x^5-18x^4-24x^3
Student: Yes
Tutor: Now if we notice, all the three terms have (-3X^3) common...
Tutor: So we'll take it out common outside the brackets...
Tutor: Any questions how we got this?
Student: No i understand i just always want to make 2 bracket instead of one
Student: That is the furthest we can factor right?
Tutor: Like this?
Tutor: Now we can further solve this expression...
Tutor: Here we have split up 6X as 4X + 2X
Tutor: This is the answer... :)
Tutor: Any questions regarding the solution?
Student: Ok, i think i just miss a step and get confused
Tutor: Which step did you miss?
Student: Next is 2x^3-x^2-8x+4=0 solve for x
Student: Factoring the 6x
Tutor: Ok :)
Student: You got it
Tutor: Taking X^2 common out of the first 2 terms and (-4) out of the last 2 terms, we get...
Tutor: Now for this product to be equal to zero, either of the brackets is equal to zero....
Tutor: Any questions so far?
Student: I would have thought x^2 . 2x =2x^3
Tutor: Yeah that is true...
Tutor: That is why when we take X^2 common, 2X remains inside the bracket...
Tutor: These are the values of X
Tutor: X has three values... +2,-2, 1/2
Tutor: Are there any more questions that I can help you with?
Student: So x= 2,-2,1/2?
Tutor: Correct!
Tutor: Yes
Student: Now i just want to check my answers: write in lowest terms:
Student: 12x^2-24x divided by 6x^2-6x-12
Tutor: Shall i solve it for you or you only want the answer?
Student: If its quicker for you to just solve that fine
Tutor: I'll solve it for you :)
Tutor: Taking 12 X common in the numerator and 6 common in the denominator...
Student: I got 2x^2-4x-12
Tutor: Close...but the answer is something else...let me show you...
Tutor: This is the answer...
Tutor: :)
Tutor: Any questions regarding the solution?
Student: No
Student: How about divide & simplify:
Tutor: Are there any more questions that I can help you with?
Tutor: Ok...
Student: X^2+5x over x^2 divided by x^2 over x-5
Student: Sorry on the first one it should be x^2 -25
Tutor: Is it correct now?
Student: No it should be x^2+5x over x^2-25 and the rest is right
Tutor: Ok...
Student: Got it
Tutor: This is the answer... :)
Tutor: Are there any more questions that I can help you with?
Student: This next one seem easy but i cant understand it 5 over p-3 subtracted by 2 over 3-p and it says to add
Tutor: Is this the question?
Student: Yes
Tutor: Ok..
Tutor: Notice that if we reverse the denominator of the second term, we'll have same denominator in both the terms...
Tutor: To reverse the denominator, we need to make it negative...
Tutor: Any questions so far?
Student: Nope
Tutor: This is the answer :)
Student: Great thats easy
Tutor: Are there any more questions that I can help you with?
Student: Now to subtract: 2x over x^2-1 subtracted by 5 over x^2-x-2
Tutor: Is this right?
Student: Yes
Tutor: Here we have factorized both the denominators...
Tutor: Any questions so far?
Student: No
Tutor: This is the answer... :)
Tutor: Are there any more questions that I can help you with?
Student: So the x+1 and the x-1 cant cancel because they are both denominators right?
Tutor: Yeah they can't cancel...
Student: How about factor completely: 6x^3 + 5x^2 -4x
Student: Are you tired yet?
Tutor: I'm not tired!! :)
Tutor: Taking X common...
Tutor: Now writing 5X = 8X - 3X
Tutor: This will be the answer... :)
Tutor: Are there any more questions that I can help you with?
Student: How did you get 8x?
Tutor: 5X = 8X - 3X
Tutor: See the quadratic awas 6X^2 + 5X - 4
Student: Yes
Tutor: We first find the product of the coefficient of X^2 and the constant
Tutor: The coefficient is 6
Tutor: And the constant term is (-4)
Tutor: So the product is (-24)
Tutor: Now we need to find two such factors of 24 whose difference is 5
Student: Ok
Student: Ok next?
Tutor: Now 24 = 1 x 24
2 x 12
3 x 8
4 x 6
Tutor: Out of these...which are the two factors whose difference is 5?
Student: 4x6?
Tutor: No... the difference between the 2 factors... like in 4 x 6 ...the difference is 2
Tutor: 6-4 =2
Tutor: The factors whose difference is 5 are 3 x 8
Student: Ok so it would be 8x3
Tutor: Because 8-3 =5
Tutor: Therefore we have splitted 5X = 8X - 3X
Student: Ok i get it now
Tutor: For example take the previous question...
Tutor: In which we had to factorize X^2 - X -2
Tutor: Here the product of the coefficient of X^2 is 1 and the constant is (-2)
Tutor: So the product is -2
Tutor: 1 x (-2) = -2
Tutor: Now we have to find two such factors of 2 whose difference is 1
Tutor: Since the coefficient of X is 1
Tutor: now 2 = 1 x 2
Tutor: That is the only possible combination...
Tutor: And this is the one which we require since 2 - 1 = 1
Tutor: Therefore we split -X = -2X + X
Tutor: Now do you get it? how we split the middle term ?
Student: Yes
Tutor: Great!! :)
Tutor: Are there any more questions that I can help you with?
Student: Yes the next question is 7x-4 over 5x + 4 over x = 9over 5 solve for x
Tutor: Is this right?
Student: No, 7x -4 over 5x then + 4/x = 9/5
Tutor: OK...
Student: The 4 goes on the bottom over the x
Student: Yes
Tutor: First we'll make the denominator common...
Tutor: Removing the denominator from the whole equation, we get...
Tutor: This is the answer :)
Tutor: Are there any more questions that I can help you with?
Student: That should do it, thanks for all of your help!!
Tutor: You are always welcome :)
Tutor: Good bye. Thank you for using Instant Math Help.