GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 19 Oct 2018, 15:00

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# How many intersects with x-axis does y = x^2 + 2qx + r have ?

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 50002
How many intersects with x-axis does y = x^2 + 2qx + r have ?  [#permalink]

### Show Tags

29 Mar 2017, 05:31
1
12
00:00

Difficulty:

65% (hard)

Question Stats:

54% (01:41) correct 46% (01:45) wrong based on 153 sessions

### HideShow timer Statistics

How many intersects with x-axis does $$y = x^2 + 2qx + r$$ have ?

(1) $$q^2 > r$$

(2) $$r^2 > q$$

_________________
SC Moderator
Joined: 13 Apr 2015
Posts: 1693
Location: India
Concentration: Strategy, General Management
GMAT 1: 200 Q1 V1
GPA: 4
WE: Analyst (Retail)
Re: How many intersects with x-axis does y = x^2 + 2qx + r have ?  [#permalink]

### Show Tags

29 Mar 2017, 20:51
2
We have to check for the value of the discriminant, b^2 - 4ac

a = 1; b = 2q; c = r
b^2 - 4ac = 4q^2 - 4r

If 4q^2 - 4r > 0 --> There are 2 intercepts for x.
4q^2 - 4r = 0 --> One x intercept
4q^2 - 4r < 0 --> No real roots

St1: q^2 > r --> 4q^2 - 4r > 0
Sufficient

St2: r^2 > q --> r can be positive or negative and we do not know whether q^2 is going to be greater or lesser than r.
Not Sufficient

Senior Manager
Joined: 06 Dec 2016
Posts: 250
Re: How many intersects with x-axis does y = x^2 + 2qx + r have ?  [#permalink]

### Show Tags

30 Mar 2017, 18:28
I do not even know what the question is asking. Can someone please explain this to me.
Manager
Joined: 02 Feb 2016
Posts: 89
GMAT 1: 690 Q43 V41
Re: How many intersects with x-axis does y = x^2 + 2qx + r have ?  [#permalink]

### Show Tags

08 Sep 2017, 15:03
This question deserves a better explanation. Anyone who could offer some help?

Bunuel
abhimahna
Board of Directors
Status: Stepping into my 10 years long dream
Joined: 18 Jul 2015
Posts: 3622
How many intersects with x-axis does y = x^2 + 2qx + r have ?  [#permalink]

### Show Tags

09 Sep 2017, 02:06
1
1
TheMastermind wrote:
This question deserves a better explanation. Anyone who could offer some help?

Hi TheMastermind ,

Here I go.

For this question, you need to understand two things:

1. If the equation is intersecting x axis, then y must be zero. Or I can say $$x^2$$ + 2qx + r = 0
2. Roots of a quadratic equation($$ax^2 + bx + c = 0$$) are given by the formula:

x = $$[ -b + \sqrt{b^2 - 4ac}]/2a$$

and

x = $$[ -b - \sqrt{b^2 - 4ac}]/2a$$

Now, in order to have real roots, the values inside the square root MUST be positive.

or I can say $$b^2 - 4ac > = 0$$

Thus, when you make the similar equation with the question in hand,you will say you need $$q^2 - r$$.

To get the real roots you will say $$q^2 >= r$$.

This is what option A is doing.

if we have $$q^2 > r$$, we will have 2 roots.

if we have $$q^2 = r$$, we will have 1 root.

Hence, A is sufficient. It tells us that we have two roots or two intersecting points.

Does that make sense?
_________________

My GMAT Story: From V21 to V40
My MBA Journey: My 10 years long MBA Dream
My Secret Hacks: Best way to use GMATClub | Importance of an Error Log!
Verbal Resources: All SC Resources at one place | All CR Resources at one place

GMAT Club Inbuilt Error Log Functionality - View More.
New Visa Forum - Ask all your Visa Related Questions - here.

Find a bug in the new email templates and get rewarded with 2 weeks of GMATClub Tests for free

Check our new About Us Page here.

Intern
Joined: 15 Mar 2017
Posts: 42
Location: India
GMAT 1: 720 Q50 V37
GPA: 4
Re: How many intersects with x-axis does y = x^2 + 2qx + r have ?  [#permalink]

### Show Tags

14 Sep 2017, 10:28
Here are my 2 cents:
The given equation is quadratic. There can be 3 cases:
2 distinct roots- graph will intersect x axis twice.
2 equal roots- graph will intersect x axis once.
2 imaginary roots - graph will not intersect x axis.

To find the nature of roots we need to look at the discriminant of the equation:

D = $$4q^2 - 4r$$

Now looking at the condition we see that $$q^2 > r$$ Hence $$q^2 - r>0$$ or $$4q^2 - 4r > 0$$
Hence D>0 and graph will intersect X axis twice.

Condition 2:
$$r^2>q$$. With this we cannot guess if $$q^2 - r>0 or <0$$
Not sufficient
_________________

You give kudos, you get kudos. :D

Senior Manager
Joined: 02 Apr 2014
Posts: 471
GMAT 1: 700 Q50 V34
How many intersects with x-axis does y = x^2 + 2qx + r have ?  [#permalink]

### Show Tags

18 Oct 2017, 13:17
The given equation points of intersection with x-axi => Basically the question is asking the number of roots of the given quadratic equation.

Case 1: Usually a quadratic equation has two roots, when it is of the form ax^2 + bx + c =0.=> no of points of intersection with x-axis is 2
Case 2: But a quadratic eqn can have both the roots equal, when it is of the form (x + a)^2 = 0 => no of points of intersection with x-axis is 1

Now, our job is to find out if the given equation falls into case2, if so no of point of intersection will be 1 else 2.
in the given equation, x^2+2qx+r = 0, if r = q^2, then equation becomes => x ^ 2 + 2qx + q ^2 => (x + q)^2 = 0 => case 2

so the question is r = q^2?

Statement 1:
q^2 > r => crystal clear sufficient that r != q^2, so the number of points of intersection is 2.

Statement 2:
r^2 > q => Not sufficient,
if q = 2, r = 4 (r = q^2), statement 2 satisfied => case 2, only one point of intersection
if q = 2, r = 5 (r != q^2), statement 2 satisfied => case 1, two points of intersection

Manager
Joined: 19 Aug 2016
Posts: 86
Re: How many intersects with x-axis does y = x^2 + 2qx + r have ?  [#permalink]

### Show Tags

21 Oct 2017, 02:46
abhimahna wrote:
TheMastermind wrote:
This question deserves a better explanation. Anyone who could offer some help?

Hi TheMastermind ,

Here I go.

For this question, you need to understand two things:

1. If the equation is intersecting x axis, then y must be zero. Or I can say $$x^2$$ + 2qx + r = 0
2. Roots of a quadratic equation($$ax^2 + bx + c = 0$$) are given by the formula:

x = $$[ -b + \sqrt{b^2 - 4ac}]/2a$$

and

x = $$[ -b - \sqrt{b^2 - 4ac}]/2a$$

Now, in order to have real roots, the values inside the square root MUST be positive.

or I can say $$b^2 - 4ac > = 0$$

Thus, when you make the similar equation with the question in hand,you will say you need $$q^2 - r$$.

To get the real roots you will say $$q^2 >= r$$.

This is what option A is doing.

if we have $$q^2 > r$$, we will have 2 roots.

if we have $$q^2 = r$$, we will have 1 root.

Hence, A is sufficient. It tells us that we have two roots or two intersecting points.

Does that make sense?

How did u conclude that q^2 > r will give two roots?

and q^2=r will give one root?

Pls help thank u
Board of Directors
Status: Stepping into my 10 years long dream
Joined: 18 Jul 2015
Posts: 3622
How many intersects with x-axis does y = x^2 + 2qx + r have ?  [#permalink]

### Show Tags

21 Oct 2017, 05:04
zanaik89 wrote:
How did u conclude that q^2 > r will give two roots?

and q^2=r will give one root?

Pls help thank u

Hi zanaik89 ,

Look at the general equations I mentioned:

x = $$[ -b + \sqrt{b^2 - 4ac}]/2a$$

and

x = $$[ -b - \sqrt{b^2 - 4ac}]/2a$$

Now, if $$\sqrt{b^2 - 4ac}$$ = 0, we will have the same value of x for both the equations.

For this to be zero, I can say $$b^2 - 4ac$$ needs to be 0. This means $$b^2$$ = 4ac

Similarly, if I get $$\sqrt{b^2 - 4ac}$$ > 0, I will get two different values of x. This means $$b^2$$ > 4ac

Use the question in hand in a similar way, you will understand entire logic.
_________________

My GMAT Story: From V21 to V40
My MBA Journey: My 10 years long MBA Dream
My Secret Hacks: Best way to use GMATClub | Importance of an Error Log!
Verbal Resources: All SC Resources at one place | All CR Resources at one place

GMAT Club Inbuilt Error Log Functionality - View More.
New Visa Forum - Ask all your Visa Related Questions - here.

Find a bug in the new email templates and get rewarded with 2 weeks of GMATClub Tests for free

Check our new About Us Page here.

How many intersects with x-axis does y = x^2 + 2qx + r have ? &nbs [#permalink] 21 Oct 2017, 05:04
Display posts from previous: Sort by