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e-GMAT Representative V
Joined: 04 Jan 2015
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What is the product of all values of x that satisfies the equation:  [#permalink]

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22 00:00

Difficulty:   85% (hard)

Question Stats: 74% (02:36) correct 26% (02:46) wrong based on 350 sessions

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What is the product of all values of x that satisfies the equation: $$\sqrt{5x} + 1= \sqrt{(7x - 3)}$$ ?

A. 9
B. 11
C. 49
D. 99
E. None of the above

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Originally posted by EgmatQuantExpert on 30 Jul 2017, 13:32.
Last edited by EgmatQuantExpert on 07 Aug 2018, 06:20, edited 2 times in total.
e-GMAT Representative V
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Posts: 3024
Re: What is the product of all values of x that satisfies the equation:  [#permalink]

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1
1
Reserving this space to post the official solution. _________________
Retired Moderator V
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Re: What is the product of all values of x that satisfies the equation:  [#permalink]

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EgmatQuantExpert wrote:
What is the product of all values of x that satisfies the equation: $$\sqrt{5x} + 1= \sqrt{(7x - 3)}$$ ?

A. 9
B. 11
C. 49
D. 99
E. None of the above

First, make sure that $$\sqrt{5x}$$ and $$\sqrt{7x-3}$$ have real value so $$x \geq 0$$ and $$x \geq \frac{3}{7}$$. Hence $$x \geq \frac{3}{7}$$.

Now, solve that equation
\begin{align} \quad \sqrt{5x}+1 &= \sqrt{7x-3} \\ 5x + 2\sqrt{5x}+1 &= 7x-3 \\ 2\sqrt{5x} &= 2x-4 \\ \sqrt{5x} &= x-2 \\ x - \sqrt{5x} - 2 &=0 \end{align}

Set $$t = \sqrt{x} \geq \sqrt{3/7}$$ we have $$t^2 -t\sqrt{5} -2 =0$$

$$\delta = 5 + 4*2 = 13 \implies t_{12}=\frac{\sqrt{5} \pm \sqrt{13}}{2}$$

We have $$t_1 = \frac{\sqrt{5}+ \sqrt{13}}{2} \approx 2.9 > 1 > \sqrt{\frac{3}{7}}$$. Choose this root.
$$t_2 = \frac{\sqrt{5}- \sqrt{13}}{2} \approx -0.7 < 0 < \sqrt{\frac{3}{7}}$$. Eliminate this root.

Hence $$x=t_1^2=\frac{9+\sqrt{65}}{2}$$.

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Joined: 26 Jan 2017
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Re: What is the product of all values of x that satisfies the equation:  [#permalink]

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The final equation i am getting is x^2-9x+4
This means no solution. EgmatQuantExpert please shed some light on the solution.
Senior Manager  S
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Re: What is the product of all values of x that satisfies the equation:  [#permalink]

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i am getting x^2-9x+4=0 which has no specific solution so answer should be E
Manager  B
Joined: 12 Sep 2016
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Re: What is the product of all values of x that satisfies the equation:  [#permalink]

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x^2-9x+4 is the final equation I get.
Is the answer E?
Current Student B
Joined: 20 Jan 2017
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Re: What is the product of all values of x that satisfies the equation:  [#permalink]

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Why don't you calculate the Discriminant.

The answer is obvious A (9)

As most of you got the final equation, x^2-9x+4=0, this needs further modification.

In equations ax^2 + bx + c = 0, the roots can be found with Discriminant.

D = b^2 - 4ac

then, x1 = ((-b) - sqrt(D))/2
x2 = ((-b) + sqrt(D))/2

So in our example

D = 81 - 16 = 65

X1 = (9-sqrt(65))/2
X2 = (9+sqrt(65))/2

Summing these two will eliminate the sqrt parts, and we will have 18/2 = 9
Senior Manager  G
Joined: 24 Apr 2016
Posts: 324
Re: What is the product of all values of x that satisfies the equation:  [#permalink]

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VachePBH wrote:
Why don't you calculate the Discriminant.

The answer is obvious A (9)

As most of you got the final equation, x^2-9x+4=0, this needs further modification.

In equations ax^2 + bx + c = 0, the roots can be found with Discriminant.

D = b^2 - 4ac

then, x1 = ((-b) - sqrt(D))/2
x2 = ((-b) + sqrt(D))/2

So in our example

D = 81 - 16 = 65

X1 = (9-sqrt(65))/2
X2 = (9+sqrt(65))/2

Summing these two will eliminate the sqrt parts, and we will have 18/2 = 9

What you have calculated appears to be the sum of the roots. Question asks for the Product of the roots.
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Re: What is the product of all values of x that satisfies the equation:  [#permalink]

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broall wrote:
EgmatQuantExpert wrote:
What is the product of all values of x that satisfies the equation: $$\sqrt{5x} + 1= \sqrt{(7x - 3)}$$ ?

A. 9
B. 11
C. 49
D. 99
E. None of the above

First, make sure that $$\sqrt{5x}$$ and $$\sqrt{7x-3}$$ have real value so $$x \geq 0$$ and $$x \geq \frac{3}{7}$$. Hence $$x \geq \frac{3}{7}$$.

Now, solve that equation
\begin{align} \quad \sqrt{5x}+1 &= \sqrt{7x-3} \\ 5x + 2\sqrt{5x}+1 &= 7x-3 \\ 2\sqrt{5x} &= 2x-4 \\ \sqrt{5x} &= x-2 \\ x - \sqrt{5x} - 2 &=0 \end{align}

Set $$t = \sqrt{x} \geq \sqrt{3/7}$$ we have $$t^2 -t\sqrt{5} -2 =0$$

$$\delta = 5 + 4*2 = 13 \implies t_{12}=\frac{\sqrt{5} \pm \sqrt{13}}{2}$$

We have $$t_1 = \frac{\sqrt{5}+ \sqrt{13}}{2} \approx 2.9 > 1 > \sqrt{\frac{3}{7}}$$. Choose this root.
$$t_2 = \frac{\sqrt{5}- \sqrt{13}}{2} \approx -0.7 < 0 < \sqrt{\frac{3}{7}}$$. Eliminate this root.

Hence $$x=t_1^2=\frac{9+\sqrt{65}}{2}$$.

Did in the same way, except for the last part. t~2,9, x=t^2, therefore x~9. Just one root -> ans. is A
Senior Manager  P
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Re: What is the product of all values of x that satisfies the equation:  [#permalink]

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I am getting E as my final answer. After squaring both sides, I get x^2-9x+4. The product is 4 which is E in our case. I am confused as to how A is the correct answer? Could anyone please explain it to me?
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Re: What is the product of all values of x that satisfies the equation:  [#permalink]

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EgmatQuantExpert wrote:
Reserving this space to post the official solution. :)

Hi eGMAT,

Can you please elaborate on the answer?
Manager  B
Joined: 19 Aug 2015
Posts: 84
Location: India
GMAT 1: 650 Q49 V30 Re: What is the product of all values of x that satisfies the equation:  [#permalink]

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As everyone solved, I am getting final equation as $$x^2-9x+4=0$$
This gives product of roots as 4 thus E is answer as further solving above equation gives roots which are slighly more than 3/7.
Manager  B
Joined: 02 Nov 2015
Posts: 163
GMAT 1: 640 Q49 V29 Re: What is the product of all values of x that satisfies the equation:  [#permalink]

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1
Probably it must be E , not A.
It will be A only if the question asked about sum of roots.

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Intern  Joined: 21 Jun 2017
Posts: 1
Re: What is the product of all values of x that satisfies the equation:  [#permalink]

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The Answer provided is wrong!
it should be E and not A

EgmatQuantExpert - Kindly Rectify this
Manager  B
Joined: 24 Jun 2017
Posts: 118
Re: What is the product of all values of x that satisfies the equation:  [#permalink]

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broall wrote:
EgmatQuantExpert wrote:
What is the product of all values of x that satisfies the equation: $$\sqrt{5x} + 1= \sqrt{(7x - 3)}$$ ?

A. 9
B. 11
C. 49
D. 99
E. None of the above

First, make sure that $$\sqrt{5x}$$ and $$\sqrt{7x-3}$$ have real value so $$x \geq 0$$ and $$x \geq \frac{3}{7}$$. Hence $$x \geq \frac{3}{7}$$.

Now, solve that equation
\begin{align} \quad \sqrt{5x}+1 &= \sqrt{7x-3} \\ 5x + 2\sqrt{5x}+1 &= 7x-3 \\ 2\sqrt{5x} &= 2x-4 \\ \sqrt{5x} &= x-2 \\ x - \sqrt{5x} - 2 &=0 \end{align}

Set $$t = \sqrt{x} \geq \sqrt{3/7}$$ we have $$t^2 -t\sqrt{5} -2 =0$$

$$\delta = 5 + 4*2 = 13 \implies t_{12}=\frac{\sqrt{5} \pm \sqrt{13}}{2}$$

We have $$t_1 = \frac{\sqrt{5}+ \sqrt{13}}{2} \approx 2.9 > 1 > \sqrt{\frac{3}{7}}$$. Choose this root.
$$t_2 = \frac{\sqrt{5}- \sqrt{13}}{2} \approx -0.7 < 0 < \sqrt{\frac{3}{7}}$$. Eliminate this root.

Hence $$x=t_1^2=\frac{9+\sqrt{65}}{2}$$.

It's actually
x=(9+√65)/2
and
x=(9-√65)/2
There are 2 solutions

the best way to check it is b^2-4ac
4 possible scenarios, one of them if the result is positive non-perfect square number then there are 2 irrational solutions but the product of them will be rational number, in that case 4
(9+√65)/2*(9-√65)/2=(81-65)/4=4

the question wasn't formulated right
I've spent like 3 or 4 minutes double and cross checking myself, 'cause i got the solution here it's but not among the available answers...
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Re: What is the product of all values of x that satisfies the equation:  [#permalink]

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pulkit0102 wrote:
The Answer provided is wrong!
it should be E and not A

EgmatQuantExpert - Kindly Rectify this

The answer is E only but I am not happy with the quality of this question. It is very not a GMAT like question.

Experts ... kindly clarify!!
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Re: What is the product of all values of x that satisfies the equation:  [#permalink]

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CAMANISHPARMAR wrote:
pulkit0102 wrote:
The Answer provided is wrong!
it should be E and not A

EgmatQuantExpert - Kindly Rectify this

The answer is E only but I am not happy with the quality of this question. It is very not a GMAT like question.

Experts ... kindly clarify!!

Hi CAMANISHPARMAR

I think you are trying to calculate the roots of the equation, which is actually not required here.

you simply need to square both sides of the equation twice to get $$x^2-9x+4$$

Now Product of roots $$= \frac{c}{a}$$, hence $$=\frac{4}{1}=4$$

This concept is frequently tested in GMAT and this question is actually very simple and can be solved under 2 minutes. Re: What is the product of all values of x that satisfies the equation:   [#permalink] 19 Jun 2018, 09:27
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