Stumped on PSAT math questions. Can anyone help?
 

My daughter's pretty good at math, but these two questions on her practice PSAT have stumped her. I'm way beyond stumped. Can anyone help... with a bit of explanation? TIA

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2x² + 7x - 15 = 0

If r and s are two solutions of the equation above and r>s, which of the following is the value of r - s ?

A) 15/2

B) 13/2

C) 11/2

D) 3/2

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And...

If x - 2 is a factor of x² - bx + b, where b is a constant, what is the value of b ?

Studying for gmat. Most equations with a number larger than 1 at the start are out of my league but I'll give it a shot. I had to do some searching on factoring this way but I need the practice. I don't see many of that type if at all on the test...

First, refer to this article https://www.mathsisfun.com/algebra/factoring-quadratics.html

I'll edit my post to show my work but it is good to read before I finish.

2x² + 7x - 15 = 0

1) Multiply 2 and -15 = -30
2) Sum of 7
--- Find set of numbers that multiply to -30 and add to 7
3) 10 and -3
4) Plug in for 7x.
5) 2x² + 10x - 3x - 15
6) Factor: 2x (x + 5) - 3 (x + 5)
7) Common factor is x + 5
8) Group -> (2x-3)(x+5)
9) Roots are 3/2 and -5
10) R > S thus 3/2 = R and -5 = S
11) R - S = 3/2 - (-5) = 3/2 + 10/2 = 13/2

Do we have an answer key for the second question?? I'm getting b = -4

PS: Thanks for the question it cleared up some things about factoring quadratics :) Easy once you know the trick

Thanks!

She says she understands the stuff in the link, but was stumped by why you were multiplying 2 and -15 in the first step.

I'm not sure about the second question though. Ask her if this seems right...


Because x - 2 is a factor, 2 = x

Plug in 2 for x and solve the linear equation. Gives you -4

Is she's good at math, have her look at this solution.
This is the way she should learn it
http://i59.tinypic.com/kdsc2c.jpg

How do you know where to put the signs?

What Sugarwood said.

factors of 15 are (1,15) or (3,5). The 2 at the beginning is going to be multiplied by one of the others.

2*1 and 1*15, with either of those being negative, there's no way to add them together and get 7. (same goes for 2*15 and 1*1)

2*3 and 1*5, with either one of these being negative and added, there's no way to get a -7, but, if you swap the 3 and 5...

2*-5 and 1*3, you get -10 and +3 which will give you -7.

that's the mental process for factoring this sort of equation. Once it's factored, you end up with what Sugarwood said.

mental trial and error

the leading interger 2 must be factored (2 and 1, 2*1 = 2). The -15 must be factored, (options are -1,15, 1,-15, -3,5 or 3,-5)

You then have to multiply and then add to get the middle interger (+7)

so

(2*-3) + (1*5) = -1

(2*3) + (1*-5) = 1

(2*-5) + (1*3) = -7 (close, but not quite)

(2*5) + (1*-3) = 10 + -3 = 7 --> (+7x)

Excellent info! If I get one of these on test day I'll nail it. Test is on October 27th.

Sorry, missed the second question. Easiest way to solve this is mental trial and error.

x^2 - bx +b

(x - 2) is one factor. b = 4

(x - 2)(x - 2)

x^2 - 2x - 2x +4 ---> x^2 - 4x + 4

For all ax^2 + bx + c = 0,

http://9f1780.medialib.glogster.com/...ic-formula.jpg

Answer is c.

(-b+/- sqrt(b*b-4ac))/2a

(-7+/-sqrt(7*7+4*2*15))/2*2

(-7+/-sqrt(49+120))/4 , sqrt (169)=13

(-7+13)/4=1.5
(-7-13)/4=-5

r>s so

r=1.5
s=-5

What is r-s?
1.5-(-5)=6.5=13/2

If x - 2 is a factor of x² - bx + b, where b is a constant, what is the value of b ?

(x-2)*(x+a)

x^2+(a-2)x+(-2a)
-a+2=-2a
2=-a -> a=-2
b=4
x^2-4x+4

Thank for all the help!

Remind me not to go back to school for any reason. My head hurts just reading that.

No kidding. Been a LONG time since I've done that stuff. Fortunately, my daughter is pretty good at math. She just hasn't had the lesson on these problems yet. I'm sure once she has, she'll do fine.

She's just taking the PSAT tomorrow as a sophomore. But, with some of her aspirations, she needs to do well on everything going forward.

I started taking the ACT as a sophomore too. It did pay off when I got to college. I even took calculus in college, but that stuff doesn't ring any bells.


I dread when the boys get into harder math, especially with all the common core crap, I'm already lost!

Lucky it foils out. Quadratic if it wanted to be difficult.
Least there isn't any trig functions to throw at it.

Good luck to your daughter!

My son is a high school senior. I've spent the last 3 years intensely studying the ACT/SAT, and the college admission process in general. It became a fairly serious hobby for me. I find it fascinating. I've read dozens of books and studies, contacted experts, conducted experiments, etc. Strange!

I have all sorts of thoughts on the (for example, are they "biased?"), but one thing is sure: They are very coachable and learnable.

That is not to say that with enough coaching and effort, anyone can get a 36/2400 (because they can't), but scores can be maximized and moved up, significantly, for anyone, through a variety of methods.

The other thing is the importance of these tests, for anyone aspiring to go to an elite or highly selective college, cannot be overstated. For highly selective colleges*, it's the single most important factor on the college application.

With a lot of thought and 2 years of hard work, I was able to help my son raise his ACT 5 points, which put his score in the top 1% and made him academically eligible at every college in the country. (He's also a recruited athlete so he had his grades and test scores "pre-read" this summer/fall by the admissions departments at half of the Top 10 schools on the Forbes America's Top Colleges list. He ended up getting offers to most of them, and others, and committed to one of them this week).

Taking the PSAT as a sophomore is a good start if she has high academic ambitions. Hopefully she does well. But if not, don't worry too much. There is still plenty of time, if you use it wisely. My son also took the PSAT as a sophomore and didn't do very well (probably around 75th percentile or so). With the right training, big improvements are pretty much guaranteed.

Let me know if you have any questions or would like any additional info, I'd love to help maximize her score.



*(The good thing, though, is of the approx. 4,000 colleges in the US, only around 30 or so are "highly selective" - admission rates of less than 15% - maybe 200 out of 4,000 are moderately selective, and 3,800 out of 4,000 are not very selective at all - 50-70%+ acceptance rates. So there's a college for everyone).

Lee, sounds like your questions have been answered.

McLovin: I am so glad to see this post. I read it off to my son and husband to illustrate that I am not the only one. I have thrown myself into the college process like this. The family jokes that I have my son's entire life planned out. My son is a sophomore so a few more years to obsess.;)

Serious question for you two....when all is said and done, does it "REALLY" matter once one hits the real world? I dunno...imo the cream will rise to the top and both of your kids are gonna be extremely successful regardless of SAT scores and the college they attend, but that's just my observation over time.

ps: You don't have to play in the SEC to be a great football player and have a stellar career...YMMV :p

Does it matter after college what the score was? No, but it does matter in getting into the university that will provide your major. If your child wants to major in Liberal Arts of some sort, there are lots of options in universities. In our case: my son is saying he wants Biomedical Engineering/PreMed. There are a handful of those. I will help any way I can to help him get into the program of his choice.

Please don't take this wrong, I'm just playing devil's advocate here ;)

For someone going into PreMed for example, does it really matter if their undergrad is done at Duke, Wake Forest, Davidson, Campbell or USC? All are awesome imo, and a star performer is gonna write their own ticket regardless of choice...just my .02.

I just think that sometimes the focus on getting into the "prestigious" schools is somewhat overrated....YMMV.

I have a relative who just entered med school at Wake....undergrad at Campbell, and he's gonna write his own ticket.

Another did hers via Harvard/Stanford (masters)....same.

Wouldn't matter where those kids did their undergrad imo....they've got "IT" SmileWavy

It's been a while. I was really lost until BlueSky posted the quadratic formula. I know it by heart, but it's been so long since I've done any of this that I didn't recognize that it applied here.

Some scholarships are based on SAT or PSAT scores. Being a National Merit Scholar can help reduce tuition costs.

Here in SC a student can get up to $10,000 per year for tuition at a state school for scoring either 1200 on SAT or 28 on ACT. That would pretty much cover tuition for any school here.

As far as schools are concerned: We want an engineering school that has Biomedical Engineering and the cost is low. With a plan of an undergrad and a medical school we want to keep to a budget.

She's aiming toward an ROTC scholarship or possibly Air Force or Naval Academies. This time next year (her junior year), she'll be in the middle of the application process. My understanding is that the PSAT is a little easier and doesn't really count, but it'll give her practice on taking this sort of test and, like in the case of these problems, show areas where she might need to do some work.

Good luck on the military academies. Their acceptance rates are some of the lowest of all universities: Somewhere around 15% rate. That would make your daughter the cream of the crop but then again not surprising from you.

My daughter's got the brains (thanks to her mom :) ) and if her drive continues to increase as it has in the past year, she's got a chance. I'd prefer the ROTC scholarship as it would be less intense than a path through the academies, but she does well when pushed. The whole ROTC thing was a bit of a surprise to my wife and me, but our daughter has really taken to the whole environment. She even signed up for her unit's Raider's team (competitive physical/mental challenges) and was made XO of her team. We'll see. Ultimate goal is flight training so she can be a pilot, liker her mom.

For an Academy, you need to secure an appointment. This is more important than grades.

My daughter (Froshman in HS) is taking Honors Geometry this year. She came home with some questions one day. I looked at it, told her Geometry was the only math class I really hated and did crappy in, and "sorry, kid. I don't remember how to do this crap." It's kinda humbling admitting to your kids you don't know everything...

To the OP...same advice I gave my daughter: there's some dude on Youtube (don't recall his name, but he's easy to find) who does GREAT tutorials on just about every type of math there is. Have her Google him.

It's one of the few formulae I still have memorized after years (and years and years) of math.

^^^^ This. More important than grades or SAT scores. Way back when, I visited the AF academy when I was 17, but just knew the military wasn't for me. Two of my hs buddies went to West Point (neither made the military their career after their obligation ended though), they didn't have the grades or SAT scores I did however, but the difference was insignificant. The point I'm trying to make is....Grades and SATs matter, but in the long run, I just don't know how much, and personally I think way too much emphasis is placed upon them....that is all ;)

For the first one factor or use the quadratic formula as the other guys have shown

for the second same idea but a little more is involved

If (x-2) is a factor of the given quadratic then the complete factorization looks like this

(x-2)(x-y) where y is the value you are looking for in order to find b

working backward by multiplying the 2 factors using the FOIL rule
x^2 -yx -2x + 2y

you have 2 true expressions to use
O + I has to sum to -bx and L has to multiply to be equal to b

both of the following are going to be true
-yx -2x =-bx because these 2 terms need to add up to the second term in the original quadratic expression, then divide by x to simplify
-y-2=-b divide by -1 and swap sides
so b=y+2

and 2y = b because the last 2 terms need to multiply to the third term in the original quadratic expression.

substituting either of these derived expressions into the other gives
2y = y+2 subtract y from both sides
y=2
substitute back into
x^2 -yx -2x + 2y
x^2 -yx -2x + 2(2)
x^2 -yx -2x + 4
so b=4 in the original question

I volunteered to proctor the PSAT at my son's school this morning. Man that was boring. The kids seemed to take it in stride. But I was in a room of juniors, so they had already taken the PSAT the previous year and most had done some practice.

My son is a sophomore, so he took the PSAT for the first time today. He did a practice test on Sunday, studied his wrong answers on Monday and Tuesday, and that was it. I'm not worried about his score this year. It's a learning experience to help him prepare for the one that counts next year.

The PSAT in your junior year matters for National Merit Scholarship consideration. Being a National Merit Scholar is a nice thing, qualifies you for some other scholarships and helps a little in college admissions.

After reading through this I don't know how I scored so high on my SATs...

Same here....rusty as hell even though I had a near perfect math score back in the day....I blame in on too many unhoppy beers and dead brain cells :p

It was a different time: I am not sure about everyone on here but when I was in High School no one took Calculus; it wasn't even offered. Now, if you are planning to go to college for engineering or hard sciences, it is almost required.

The SAT was revamped this year and prior to that in 2005 to make it harder, or so they say.

It makes it all the more surprising with the "Math is Hard" posts on here. We have greatly failed our students if we are requiring higher math like Calculus in High School but they can't do simple math in their heads.

McLovin,

For the next tier (just below Ivy and elite) schools, is it important to take both the ACT and SAT tests as a junior?

Or should the student find which one he/she is good at and concentrate on that? I heard that most schools take either one and count it equally.

Wondering what the reality is in terms of school preference.

I can't help with the math problem, (and have no idea now how I scored well on the PSAT in HS), but my nephew is an applied math major and senior in college. His stories about the math library are hilarious; undergrads are not strange but anyone grad level or older, (instructors), look like they are in a coma. He says it's like an insane asylum. :D