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By how much does the larger root of the equation 2x^2+5x = 1
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By how much does the larger root of the equation 2x^2+5x = 12 exceed the smaller root?


(A) 5/2
(B) 10/3
(C) 7/2
(D) 14/3
(E) 11/2

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Source: GMATPrep Question Pack 1
Difficulty: Hard
------------
The problem I was having with this question was factoring out

2x^2 + 5x -12 = 0

How do I factor the equation when the x^2 has a coefficient that's not 1 -- in this case 2?

Thanks
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By how much does the larger root of the equation 2x^2+5x = 1
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 Updated on: 03 Nov 2014, 00:23
Originally posted by PareshGmat on 31 Oct 2013, 00:25.
Last edited by PareshGmat on 03 Nov 2014, 00:23, edited 1 time in total.
Re-writing the equation as follows:

\(2x^2 + 5x - 12 =0\)

The two factors of -24 (-12 x 2) are 8 & -3

So,\(2x^2 + 8x - 3x - 12 = 0\)

2x (x +4) -3 (x+4) = 0

(2x -3) (x+4) = 0

So \(x = \frac{3}{2}\) or x = -4
Distance between 3/2 & -4 =

\(\frac{3}{2} - (-4) = \frac{11}{2}\) (Answer = E)
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By how much does the larger root of the equation 2x^2+5x = 1
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02 Apr 2015, 11:22
Hi All,

It looks like a number of the explanations take a more complex approach than what is needed. This question is based on FOIL-ing and Factoring rules; even though it looks a little "tough", the same rules still apply...

We're given 2X^2 + 5X = 12

We can rewrite that as....

2X^2 + 5X - 12 = 0

Now let's break this into it's two 'pieces'....

(X _ _ )(2X _ _ )

Now let's look at the '-12'....

This means that the two numbers could be....
1 and 12
2 and 6
3 and 4

And one number is + and the other is -

Since the middle term of the Quadratic is "5X", we need to 'play around' a bit with the possibilities....

1 and 12 are too far 'apart'
2 and 6 are both even, so we won't end up with 5X (since 5 is odd)

That just leaves us with 3 and 4....
(X + 4)(2X - 3) = 0

Now we can solve the Quadratic...

X = -4, +3/2

The prompt asks for the difference in the solutions...
(3/2) - (-4) = 11/2

Final Answer:
Show Spoiler
E

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By how much does the larger root of the equation 2x^2+5x = 1
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04 Aug 2013, 10:00
DelSingh
By how much does the larger root of the equation 2x^2+5x = 12 exceed the smaller root?


(A) 5/2
(B) 10/3
(C) 7/2
(D) 14/3
(E) 11/2

Source: GMATPrep Question Pack 1
Difficulty: Hard
------------
The problem I was having with this question was factoring out

2x^2 + 5x -12 = 0

How do I factor the equation when the x^2 has a coefficient that's not 1 -- in this case 2?

Thanks
Show more

You don't need to factor out anything for this question. Note that \((a-b)^2 = (a+b)^2-4ab\)

Now, the sum of the roots :\(\frac{-5}{2}\) and product of the roots :\(\frac{-12}{2}\)

Thus, \((a-b)^2 = \frac{25}{4}+4*6 = \frac{121}{4}\)

Thus, \((a-b) = \frac{11}{2}\)

E.

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By how much does the larger root of the equation 2x^2+5x = 1
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04 Aug 2013, 10:00
DelSingh
By how much does the larger root of the equation 2x^2+5x = 12 exceed the smaller root?


(A) 5/2
(B) 10/3
(C) 7/2
(D) 14/3
(E) 11/2

Source: GMATPrep Question Pack 1
Difficulty: Hard
------------
The problem I was having with this question was factoring out

2x^2 + 5x -12 = 0

How do I factor the equation when the x^2 has a coefficient that's not 1 -- in this case 2?

Thanks
Show more

TO DETERMINE THE ROOTS OF QUADRATIC EQUATION: \(ax^2+bx+c = 0\)

formula for root = \((-b+\sqrt{(b^2-4ac)})/2a\) and \((-b-\sqrt{(b^2-4ac)})/2a\)

now in your equation:\(2x^2+5x-12 = 0\)
\(a=2\\
b=5\\
c=-12\)

now when you will plug in the values in the formula

roots come out are = \(-4 and 3/2\)

subtracting smaller from bigger will give you \(11/2\)

hope it helps
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By how much does the larger root of the equation 2x^2+5x = 1
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04 Aug 2013, 10:27
mau5
DelSingh
By how much does the larger root of the equation 2x^2+5x = 12 exceed the smaller root?


(A) 5/2
(B) 10/3
(C) 7/2
(D) 14/3
(E) 11/2

Source: GMATPrep Question Pack 1
Difficulty: Hard
------------
The problem I was having with this question was factoring out

2x^2 + 5x -12 = 0

How do I factor the equation when the x^2 has a coefficient that's not 1 -- in this case 2?

Thanks

You don't need to factor out anything for this question. Note that \((a-b)^2 = (a+b)^2-4ab\)

Now, the sum of the roots :\(\frac{-5}{2}\) and product of the roots :\(\frac{-12}{2}\)

Thus, \((a-b)^2 = \frac{25}{4}+4*6 = \frac{121}{4}\)

Thus, \((a-b) = \frac{11}{2}\)

E.
Show more

Note that \((a-b)^2 = (a+b)^2-4ab\)

How are you getting this? I thought \((a-b)^2 = a^2 -2ab+ b^2\)
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By how much does the larger root of the equation 2x^2+5x = 1
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04 Aug 2013, 10:30
DelSingh
mau5
DelSingh
By how much does the larger root of the equation 2x^2+5x = 12 exceed the smaller root?


(A) 5/2
(B) 10/3
(C) 7/2
(D) 14/3
(E) 11/2

Source: GMATPrep Question Pack 1
Difficulty: Hard
------------
The problem I was having with this question was factoring out

2x^2 + 5x -12 = 0

How do I factor the equation when the x^2 has a coefficient that's not 1 -- in this case 2?

Thanks

You don't need to factor out anything for this question. Note that \((a-b)^2 = (a+b)^2-4ab\)

Now, the sum of the roots :\(\frac{-5}{2}\) and product of the roots :\(\frac{-12}{2}\)

Thus, \((a-b)^2 = \frac{25}{4}+4*6 = \frac{121}{4}\)

Thus, \((a-b) = \frac{11}{2}\)

E.

Note that \((a-b)^2 = (a+b)^2-4ab\)

How are you getting this? I thought \((a-b)^2 = a^2 -2ab+ b^2\)
Show more

\((a+b)^2-4ab = a^2+b^2+2ab-4ab = a^2 -2ab+ b^2\)

Hope this helps.

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By how much does the larger root of the equation 2x^2+5x = 1
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04 Aug 2013, 10:57
mau5

You don't need to factor out anything for this question. Note that \((a-b)^2 = (a+b)^2-4ab\)

Now, the sum of the roots :\(\frac{-5}{2}\) and product of the roots :\(\frac{-12}{2}\)

Thus, \((a-b)^2 = \frac{25}{4}+4*6 = \frac{121}{4}\)

Thus, \((a-b) = \frac{11}{2}\)

E.

Note that \((a-b)^2 = (a+b)^2-4ab\)

How are you getting this? I thought \((a-b)^2 = a^2 -2ab+ b^2\)

\((a+b)^2-4ab = a^2+b^2+2ab-4ab = a^2 -2ab+ b^2\)

Hope this helps.

\((a+b)^2-4ab = a^2+b^2+2ab-4ab = a^2 -2ab+ b^2\)
Show more

Sorry for buggin', but I am still curious as to why you chose to manipulate \((a-b)^2 into (a+b)^2-4ab\) when you encountered this problem? Is there some sort of method/property that comes to mind? The study guides I am using doesn't really show this, but I would love to know :)
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By how much does the larger root of the equation 2x^2+5x = 1
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04 Aug 2013, 11:06
DelSingh
mau5

You don't need to factor out anything for this question. Note that \((a-b)^2 = (a+b)^2-4ab\)

Now, the sum of the roots :\(\frac{-5}{2}\) and product of the roots :\(\frac{-12}{2}\)

Thus, \((a-b)^2 = \frac{25}{4}+4*6 = \frac{121}{4}\)

Thus, \((a-b) = \frac{11}{2}\)

E.

Note that \((a-b)^2 = (a+b)^2-4ab\)

How are you getting this? I thought \((a-b)^2 = a^2 -2ab+ b^2\)

\((a+b)^2-4ab = a^2+b^2+2ab-4ab = a^2 -2ab+ b^2\)

Hope this helps.

\((a+b)^2-4ab = a^2+b^2+2ab-4ab = a^2 -2ab+ b^2\)

Sorry for buggin', but I am still curious as to why you chose to manipulate \((a-b)^2 into (a+b)^2-4ab\) when you encountered this problem? Is there some sort of method/property that comes to mind? The study guides I am using doesn't really show this, but I would love to know :)
Show more

I chose this method only in this context . The question was asking for the difference of roots.
.
Now, we alrady know the sum and the product of the 2 roots. The formula which I have used is just to get the difference of the 2.

By the way , it might be a handy formula to remember.

Hope this helps.

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By how much does the larger root of the equation 2x^2+5x = 1
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04 Aug 2013, 11:07
DelSingh
mau5

You don't need to factor out anything for this question. Note that \((a-b)^2 = (a+b)^2-4ab\)

Now, the sum of the roots :\(\frac{-5}{2}\) and product of the roots :\(\frac{-12}{2}\)

Thus, \((a-b)^2 = \frac{25}{4}+4*6 = \frac{121}{4}\)

Thus, \((a-b) = \frac{11}{2}\)

E.

Note that \((a-b)^2 = (a+b)^2-4ab\)

How are you getting this? I thought \((a-b)^2 = a^2 -2ab+ b^2\)

\((a+b)^2-4ab = a^2+b^2+2ab-4ab = a^2 -2ab+ b^2\)

Hope this helps.

\((a+b)^2-4ab = a^2+b^2+2ab-4ab = a^2 -2ab+ b^2\)

Sorry for buggin', but I am still curious as to why you chose to manipulate \((a-b)^2 into (a+b)^2-4ab\) when you encountered this problem? Is there some sort of method/property that comes to mind? The study guides I am using doesn't really show this, but I would love to know :)
Show more

for a quadratic equation AX^2+BX+C = 0
SUM OF ROOTS = -B/A
PRODUCT OF ROOTS = C/A
let a AND b be the roots of equation
then a*b = C/A
a + b = -B/A

now as we have to calculate difference of roots (a-b)
we can use directly the formula (a-b)^2 = (a+b)^2 - 4ab...now simply you have to plug in the values..

hope it helps
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By how much does the larger root of the equation 2x^2+5x = 1
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01 Mar 2018, 22:07
Hi all,

For those of you who wish to understand quadratics better or are already scrambled with formulas, I suggest to anyone that struggles with quadratics to go to this site (recommended by Bunuel): https://www.purplemath.com/modules/factquad.htm

The box method mentioned on the site makes questions like this cake.
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By how much does the larger root of the equation 2x^2+5x = 1
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01 Mar 2018, 22:14
mrosale2
Hi all,

For those of you who wish to understand quadratics better or are already scrambled with formulas, I suggest to anyone that struggles with quadratics to go to this site (recommended by Bunuel): https://www.purplemath.com/modules/factquad.htm

The box method mentioned on the site makes questions like this cake.
Show more

In addition to that.

7. Algebra



For more check Ultimate GMAT Quantitative Megathread

Hope it helps.

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By how much does the larger root of the equation 2x^2+5x = 1
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20 Nov 2020, 06:30
DelSingh
By how much does the larger root of the equation 2x^2+5x = 12 exceed the smaller root?


(A) 5/2
(B) 10/3
(C) 7/2
(D) 14/3
(E) 11/2
Show more
\(2x^2+5x -12 = 0\)

Or, \(2x^2+8x - 3x -12 = 0\)

Or, \(2x( x + 4 ) - 3 ( x + 4) = 0\)

Or, \(x =\frac{3}{2}\) & \(x = -4\)

So , Larger root exceeds smaller root by \(\frac{3}{2} + 4 = \frac{11}{2}\), Answer must be (E)

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By how much does the larger root of the equation 2x^2+5x = 1
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24 Jan 2024, 22:00
roots of the equation is given by

(-b ± √D)/2a

therefore the distance between them is given by

√D/2a * 2 => √D/a

D here is given by b^2 - 4ac = 5*5 - 4(2)(-12) = 121

therefore the answer is √121/2 = 11/2

This is just to make your concepts clear....it's better to actually take the roots by the method taught in our school and then see their distance.
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By how much does the larger root of the equation 2x^2+5x = 1
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07 Oct 2025, 02:51
This is a great quadratic equation problem that tests not just your ability to find roots, but also your understanding of what the question is really asking. Let's work through it together.

Understanding what you're being asked

When the problem asks "by how much does the larger root exceed the smaller root," it's asking for the difference between the two roots. Think of it as: Larger root - Smaller root. Keep this in mind as we solve.

Step 1: Set up the equation properly

You start with \(2x^2 + 5x = 12\). To use the quadratic formula, you need to move everything to one side:

\(2x^2 + 5x - 12 = 0\)

Now you have \(a = 2\), \(b = 5\), and \(c = -12\). Watch that negative sign on the 12!

Step 2: Find both roots using the quadratic formula

The quadratic formula is: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Let's calculate what's under the square root (the discriminant):
\(b^2 - 4ac = 5^2 - 4(2)(-12) = 25 + 96 = 121\)

Notice how \(4(2)(-12) = -96\), so you're actually doing \(25 - (-96)\), which becomes \(25 + 96\).

Since \(\sqrt{121} = 11\), your formula becomes:
\(x = \frac{-5 \pm 11}{4}\)

This gives you two solutions:
  • Larger root: \(x = \frac{-5 + 11}{4} = \frac{6}{4} = \frac{3}{2}\)
  • Smaller root: \(x = \frac{-5 - 11}{4} = \frac{-16}{4} = -4\)

Step 3: Calculate the difference

Here's the key step: Larger root - Smaller root

\(\frac{3}{2} - (-4) = \frac{3}{2} + 4\)

Converting 4 to a fraction with denominator 2: \(4 = \frac{8}{2}\)

So: \(\frac{3}{2} + \frac{8}{2} = \frac{11}{2}\)

Answer: (E) \(\frac{11}{2}\)

The complete solution on Neuron breaks down the strategic approach for all quadratic root problems, highlights the common mistakes students make (especially with sign errors and fraction arithmetic), and shows you the process skills for translating problem language into the right mathematical operations. You can check out the step-by-step solution on Neuron by e-GMAT to master the systematic root-finding approach. You can also explore other GMAT official questions with detailed solutions on Neuron for structured practice here.

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