Stumped on PSAT math questions. Can anyone help?

[LeeH]

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 ?

[JD159]

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

[LeeH]

Originally Posted by JD159

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

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.

[JD159]

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

[sugarwood]

Is she's good at math, have her look at this solution.
This is the way she should learn it

[JD159]

Originally Posted by sugarwood

Is she's good at math, have her look at this solution.
This is the way she should learn it

How do you know where to put the signs?

[masraum]

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.

[masraum]

Originally Posted by JD159

How do you know where to put the signs?

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)

[JD159]

Originally Posted by masraum

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.

[masraum]

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

[BlueSkyJaunte]

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

[red-beard]

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

[red-beard]

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

[LeeH]

Thank for all the help!

[porsche4life]

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

[LeeH]

Originally Posted by porsche4life

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.

[porsche4life]

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!

[Arizona_928]

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

[McLovin]

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).

[PorscheGAL]

Originally Posted by McLovin

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.