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What is the product of all values of x that satisfies the equation:

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What is the product of all values of x that satisfies the equation: [#permalink]

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New post Updated on: 19 Sep 2017, 01:45
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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 Bunuel on 19 Sep 2017, 01:45, edited 1 time in total.
Edited the OA.
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Re: What is the product of all values of x that satisfies the equation: [#permalink]

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New post 30 Jul 2017, 13:42
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1
Reserving this space to post the official solution. :)
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Re: What is the product of all values of x that satisfies the equation: [#permalink]

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New post 31 Jul 2017, 01:51
2
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 \(x=t_1^2=\frac{9+\sqrt{65}}{2}\).

The answer E.
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Re: What is the product of all values of x that satisfies the equation: [#permalink]

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New post 31 Jul 2017, 11:38
1
The final equation i am getting is x^2-9x+4
This means no solution. EgmatQuantExpert please shed some light on the solution.
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Re: What is the product of all values of x that satisfies the equation: [#permalink]

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New post 01 Aug 2017, 23:33
i am getting x^2-9x+4=0 which has no specific solution so answer should be E
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Re: What is the product of all values of x that satisfies the equation: [#permalink]

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New post 02 Aug 2017, 05:19
x^2-9x+4 is the final equation I get.
Is the answer E?
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Re: What is the product of all values of x that satisfies the equation: [#permalink]

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New post 02 Aug 2017, 10:43
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
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Re: What is the product of all values of x that satisfies the equation: [#permalink]

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New post 02 Aug 2017, 11:20
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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New post 02 Aug 2017, 11:37
1
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 \(x=t_1^2=\frac{9+\sqrt{65}}{2}\).

The answer E.

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

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New post 03 Aug 2017, 12:28
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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New post 17 Aug 2017, 21:48
EgmatQuantExpert wrote:
Reserving this space to post the official solution. :)



Hi eGMAT,

Can you please elaborate on the answer?
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Re: What is the product of all values of x that satisfies the equation: [#permalink]

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New post 19 Aug 2017, 04:42
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.
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Re: What is the product of all values of x that satisfies the equation: [#permalink]

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New post 19 Aug 2017, 08:58
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Probably it must be E , not A.
It will be A only if the question asked about sum of roots.

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

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New post 20 Aug 2017, 01:08
The Answer provided is wrong!
it should be E and not A

EgmatQuantExpert - Kindly Rectify this
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What is the product of all values of x that satisfies the equation: [#permalink]

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New post 22 Sep 2017, 20:32
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 \(x=t_1^2=\frac{9+\sqrt{65}}{2}\).

The answer E.


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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What is the product of all values of x that satisfies the equation: [#permalink]

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New post 19 Jun 2018, 09:09
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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New post 19 Jun 2018, 09:27
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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