I'm stumped. Help me solve these two problems.

[Hot Lava Mama]

Ds's math book does not provide the answers for the even problems.  I work out the "cumulative review" even problems in every chapter because it is good review for him (and me!).  I must be too tired because I can't seem to figure these out.  Can you help?

 

Factor completely:

16  -  (x - 11) {x - 11 is squared, but I can't figure out how to write that in this forum)

 

 

Factor completely:

 

10x{the x is squared}  +  13x  -  3

 

Thanks for your help!

Hot Lava Mama

If you are just lazy and want the answers, you can type them into wolfram alpha online, and have it solve them for you, with nice graphs and everything:

 

http://www.wolframalpha.com/input/?i=16+-+%28x+-+11%29+%5E+2

 

http://www.wolframalpha.com/input/?i=10x%5E2++%2B+13x++-3

[ErinE]

Ds's math book does not provide the answers for the even problems. I work out the "cumulative review" even problems in every chapter because it is good review for him (and me!). I must be too tired because I can't seem to figure these out. Can you help?

 

Factor completely:

16 - (x - 11) {x - 11 is squared, but I can't figure out how to write that in this forum)

 

 

Factor completely:

 

10x{the x is squared} + 13x - 3

 

Thanks for your help!

Hot Lava Mama

A quick reply to the second

10x^2 + 13 x - 3

 

Factors of 10 are 1, 2, 5, 10. Factors of -3 are -3, -1, 1, 3.

 

Looking at the possible factor combinations, I know +15x (5x*3) -2x (2x*(-1)) equals 13x so the solution is:

(5x - 1)(2x +3)

 

ETA: the first

 

16 - (x-11)^2

 

Find (x-11)^2

16 - (x^2 -22x + 121)

 

Distribute the -1

16 - x^2 + 22x - 121

 

Combine like terms (16 and -121)

-x^2 + 22x - 105

 

Factor out -1

-(x^2-22x+105)

 

Factors of 105 that combine to 22 are 7 and 15 so solution is:

-(x-15)(x-7)

[Arcadia]

Factor completely:

16 - (x - 11) {x - 11 is squared, but I can't figure out how to write that in this forum)

 

16 - (x-11)^2 is the same as

4^2 - (x-11)^2

 

a^2 - b^2 = (a-b )(a+b )

4^2 - (x-11)^2 = (4-x+11)(4+x-11) where a = 4 and b = x - 11

= (15-x)(x-7)

[8filltheheart]

16 - (x-11)^2 is the same as

4^2 - (x-11)^2

 

a^2 - b^2 = (a-b )(a+b )

4^2 - (x-11)^2 = (4-x+11)(4+x-11) where a = 4 and b = x - 11

= (15-x)(x-7)

 

x = 15, x = 7

This is a great explanation. The only thing I would not want the OP to misunderstand is that you should not solve for the variables. The original problem was not given as an equation but an expression.

[Arcadia]

The only thing I would not want the OP to misunderstand is that you should not solve for the variables. The original problem was not given as an equation but an expression.

Oops, I'll edit my post.

Old habit from having to factorise, solve for x and plot the graph for full credit for a math question.

[quark]

Oops, I'll edit my post.

Old habit from having to factorise, solve for x and plot the graph for full credit for a math question.

 

OT alert: I had a teacher who required it even if the question didn't ask for it. I liked your elegant steps!

[Hot Lava Mama]

If you are just lazy and want the answers, you can type them into wolfram alpha online, and have it solve them for you, with nice graphs and everything:

 

http://www.wolframalpha.com/input/?i=16+-+%28x+-+11%29+%5E+2

 

http://www.wolframalpha.com/input/?i=10x%5E2++%2B+13x++-3

 

:)  Snicker!  No, not lazy in this case, just stupid!  :)  Thanks for that link!  I never heard of it.  I am sure it will be useful during our coming high school days!  (My oldest is just starting 9th, and I have 4 other kids behind him!)

 

Thanks!

Hot Lava Mama

[Hot Lava Mama]

A quick reply to the second

10x^2 + 13 x - 3

 

Factors of 10 are 1, 2, 5, 10. Factors of -3 are -3, -1, 1, 3.

 

Looking at the possible factor combinations, I know +15x (5x*3) -2x (2x*(-1)) equals 13x so the solution is:

(5x - 1)(2x +3)

 

ETA: the first

 

16 - (x-11)^2

 

Find (x-11)^2

16 - (x^2 -22x + 121)

 

Distribute the -1

16 - x^2 + 22x - 121

 

Combine like terms (16 and -121)

-x^2 + 22x - 105

 

Factor out -1

-(x^2-22x+105)

 

Factors of 105 that combine to 22 are 7 and 15 so solution is:

-(x-15)(x-7)

 

Thank you!  My mind it on vacation right now!  :)  I appreciate the help! (Thank your brain for me, too!)

:)

Hot Lava Mama

 

[Hot Lava Mama]

A quick reply to the second

10x^2 + 13 x - 3

 

Factors of 10 are 1, 2, 5, 10. Factors of -3 are -3, -1, 1, 3.

 

Looking at the possible factor combinations, I know +15x (5x*3) -2x (2x*(-1)) equals 13x so the solution is:

(5x - 1)(2x +3)

 

ETA: the first

 

16 - (x-11)^2

 

Find (x-11)^2

16 - (x^2 -22x + 121)

 

Distribute the -1

16 - x^2 + 22x - 121

 

Combine like terms (16 and -121)

-x^2 + 22x - 105

 

Factor out -1

-(x^2-22x+105)

 

Factors of 105 that combine to 22 are 7 and 15 so solution is:

-(x-15)(x-7)

Thanks!  That -1 on the outside of (x-15) would have tripped me up.  I was stumped with the middle piece.

 

:)

Hot Lava Mama

[Cosmos]

OT alert: I had a teacher who required it even if the question didn't ask for it. I liked your elegant steps!

So if the problem asked the student to factor an expression, the teacher expected the student also to imagine that the expression appeared in an equation and solve that imaginary equation?  I always wondered why so many of the students I tutored seemed not to understand the distinction between an expression and an equation (or between simplifying and expression and solving an equation). They were always trying to solve an equation when there wasn't one!

[ErinE]

Thanks! That -1 on the outside of (x-15) would have tripped me up. I was stumped with the middle piece.

 

:)

Hot Lava Mama

I factor out the -1 because otherwise there would be two possible combinations

 

(15-x)(x-7)

and

(7-x)(x-15)

 

Any of the three would be correct, but I would argue the most "factored" (if that's a real term) is:

-(x-15)(x-7)

[ErinE]

16 - (x-11)^2 is the same as

4^2 - (x-11)^2

 

a^2 - b^2 = (a-b )(a+b )

4^2 - (x-11)^2 = (4-x+11)(4+x-11) where a = 4 and b = x - 11

= (15-x)(x-7)

I like your solution as well. I always tell DS to be a thinker, not a computer then I find myself guilty of computing!

[quark]

So if the problem asked the student to factor an expression, the teacher expected the student also to imagine that the expression appeared in an equation and solve that imaginary equation?  I always wondered why so many of the students I tutored seemed not to understand the distinction between an expression and an equation (or between simplifying and expression and solving an equation). They were always trying to solve an equation when there wasn't one!

I'm trying to remember exactly what we did, what was expected and how it was phrased. I learned math in a different language and that is another reason why I stumble a lot with it in English although English is my more fluent language. I remember clearly having to show a lot of things not even asked for in the question and half of the time I was figuring out what exactly the teacher wanted and what the question was asking for. Not saying it was the right or wrong thing to do. Arcadia's remark brought back memories. :)

 

One of the reasons why I always maintain that I don't teach my kid upper level math. I'm still re-learning math terms in English myself.

[Arcadia]

So if the problem asked the student to factor an expression, the teacher expected the student also to imagine that the expression appeared in an equation and solve that imaginary equation? 

The problem is actually worded differently.  Below link is a typical workbook exercise question I had as a kid. Can't remember if NEM1 might have similar style questions.

http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/quadequationshirev1.shtml

[quark]

The problem is actually worded differently.  Below link is a typical workbook exercise question I had as a kid. Can't remember if NEM1 might have similar style questions.

http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/quadequationshirev1.shtml

 

Thanks for the link...does that bring back more memories! Yes, I remember having to complete that step where you find x when the two factors multiplied = 0. Still trying to find examples online. I'll update when I do.

[8filltheheart]

The problems should be worded differently. Solving for the variable requires 2 expressions to be set equal to each other. An expression is not set equal to anything. In order to solve it, you would have to make the assumption that it equalled zero.

 

The link Arcadia posted is an equation and can be solved. The expression equals zero. That site also has info on expressions. http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/factorisinghirev1.shtml. They are factored, but not solved for the variable.

[quark]

My apologies! I looked through an A Level maths book that I have at home that comes closest to the type of questions we did. I think I was confusing things. Cosmos, I was probably like one of your tutored students! :P

[Cosmos]

The problem is actually worded differently.  Below link is a typical workbook exercise question I had as a kid. Can't remember if NEM1 might have similar style questions.

http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/quadequationshirev1.shtml

 

Sure. I think that type of problem is typical of any algebra program.

 

I just think that a teacher should be cautious in his use of language to ensure that the students understand the difference between these two problems:

 

Solve the equation x^2 - 5x + 4 = 0

 

and

 

Simplify the expression x^2 - 5x + 4

 

The problem in the link clearly indicates the first type.

 

I do not think it benefits students to give the impression that the simplifying of an expression is inevitably followed by setting that expression to zero and solving for roots, even though that is of course a very common procedure to follow. So it surprises me to hear that a teacher might say "simplify an expression" when he intends "solve an equation".

[Cosmos]

I learned math in a different language and that is another reason why I stumble a lot with it in English although English is my more fluent language.

 

Isn't that an interesting thing to think about? I wonder how much influence the language of instruction has on our math understanding. You hear people talk about the influence that numbering systems ("one-ten two" instead of "twelve", for example) has on children's understanding of place value, but I haven't heard that idea explored very fully into other areas of mathematics. A bit of a side track for this thread, though!

[kiana]

I do not think it benefits students to give the impression that the simplifying of an expression is inevitably followed by setting that expression to zero and solving for roots, even though that is of course a very common procedure to follow. So it surprises me to hear that a teacher might say "simplify an expression" when he intends "solve an equation".

+1 googolplex.

 

I see people do this all the time in my classes (so it is a very common error), but if there is no equation, you cannot solve the expression. Or, as I tell my students, "If there is no 'equals' in the problem, why are you putting one in the answer?"

 

This is compounded by the use of equals to mean "and my next step is" which is another extremely common error. Equals should only be used between things that are actually equal. This hinders students when they need to reason through chains of equalities using transitivity.

 

Sorry about the continued sidetrack.

[katilac]

Slight sidetrack: check your library and see if they offer free online tutoring. Ours does, in every subject! We just enter our library card number and 'wait in line' for a tutor. They ask if you need help with an entire lesson, if you are stuck on a certain problem, etc. We've never had to wait more than a few minutes.

 

This saves us a LOT of time in math, lol. You and the tutor can type messages back and forth, and there is a whiteboard for solving problems.

 

I haven't tried this part yet, but plan to: you can submit an essay/paper for feedback!

[Dana]

Sure. I think that type of problem is typical of any algebra program.

 

I just think that a teacher should be cautious in his use of language to ensure that the students understand the difference between these two problems:

 

And that's why I think it's important that students also can explain why they're doing what they're doing (as per discussion on chat board about explanations in math). Getting the right answer doesn't mean much IMO if you can't also say why it's right. (And I'd also argue that this is a major problem with teachers who don't know their math well enough.)