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If then x =
[#permalink]
28 Oct 2018, 17:31

2

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Question Stats:

If then \(x^2 - x \sqrt{2} + 3x\sqrt{3}=\sqrt{54}\) then \(x =\)

Indicate all possible values.

A. \(-\sqrt{2}\)

B. \(-3\sqrt{2}\)

C. \(-3\sqrt{3}\)

D. \(\sqrt{2}\)

_________________

Indicate all possible values.

A. \(-\sqrt{2}\)

B. \(-3\sqrt{2}\)

C. \(-3\sqrt{3}\)

D. \(\sqrt{2}\)

_________________

Sandy

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Re: If then x =
[#permalink]
14 Dec 2018, 07:29

1

sandy wrote:

If then \(x^2 - x \sqrt{2} + 3x\sqrt{3}=\sqrt{54}\) then \(x =\)

Indicate all possible values.

A. \(-\sqrt{2}\)

B. \(-3\sqrt{2}\)

C. \(-3\sqrt{3}\)

D. \(\sqrt{2}\)

Indicate all possible values.

A. \(-\sqrt{2}\)

B. \(-3\sqrt{2}\)

C. \(-3\sqrt{3}\)

D. \(\sqrt{2}\)

One option is to plug in each answer choice and see if it satisfies the given equation.

It's time-consuming, but works.

Here's another approach:

GIVEN: x² - x√2 + 3x√3 = √54

Simplify √54

√54 = √[(9)(6)] = (√9)(√6) = 3√6

So, we can write: x² - x√2 + 3x√3 = 3√6

Subtract 3√6 from both sides to get: x² - x√2 + 3x√3 - 3√6 = 0

From here, we'll use an advanced technique called FACTORING IN PARTS.

Factor the greatest common factor from the first two terms to get: x(x - √2) + 3x√3 - 3√6 = 0

Factor the last two terms to get: x(x - √2) + 3√3(x - √2) = 0

COMBINE the terms to get: (x + 3√3)(x - √2) = 0

From here, we can see that the solutions are: x = -3√3 and x = √2

Answer: C, D

Cheers,

Brent

_________________

If then x =
[#permalink]
31 Dec 2023, 22:59

1

If \(x^2 - x \sqrt{2} + 3x\sqrt{3}=\sqrt{54}\) then \(x =\)

The moment we see something monstrously complicated as

\(x^2 - x \sqrt{2} + 3x\sqrt{3}=\sqrt{54}\)

we can be sure that there is a simple solution.

The reason being, the GRE is not a test of our calculating prowess, but a test of our alertness in spotting opportunities that enable an easy solution for the problem. The GRE rewards clever test takers by providing them opportunities to save precious time.

Right off the bat, we see that it is a quadratic equation.

so, let us begin by gathering all the like terms and writing the equation in the standard form.

\(x^2 +(3\sqrt{3} - \sqrt{2}) - \sqrt{54} = 0\)

Now, we need two numbers which when added together give us the coefficient of \(\text{x}\) and when multiplied together give us the constant term.

But we see that the coefficient of \(\text{x}\) is already a sum, consisting of the following two numbers :

\(3\sqrt{3}, -\sqrt{2}.\)

Could it be that the product of these two numbers yields\( -\sqrt{54}\text{ ? }\)

Well, lets see. \(3\sqrt{3} \times \sqrt{2} = -3\sqrt{6}\)

And \(-\sqrt{54} = -1 \times \sqrt{9 \times 6} = -3\sqrt{6 }

\)

We see that it is indeed the case. The two numbers are \(3\sqrt{3} \text{ and } -\sqrt{2}\)

Therefore,\( x = -3\sqrt{3} \text{ and } \sqrt{2}\)

The answers are Choices C and D

_________________

The moment we see something monstrously complicated as

\(x^2 - x \sqrt{2} + 3x\sqrt{3}=\sqrt{54}\)

we can be sure that there is a simple solution.

The reason being, the GRE is not a test of our calculating prowess, but a test of our alertness in spotting opportunities that enable an easy solution for the problem. The GRE rewards clever test takers by providing them opportunities to save precious time.

Right off the bat, we see that it is a quadratic equation.

so, let us begin by gathering all the like terms and writing the equation in the standard form.

\(x^2 +(3\sqrt{3} - \sqrt{2}) - \sqrt{54} = 0\)

Now, we need two numbers which when added together give us the coefficient of \(\text{x}\) and when multiplied together give us the constant term.

But we see that the coefficient of \(\text{x}\) is already a sum, consisting of the following two numbers :

\(3\sqrt{3}, -\sqrt{2}.\)

Could it be that the product of these two numbers yields\( -\sqrt{54}\text{ ? }\)

Well, lets see. \(3\sqrt{3} \times \sqrt{2} = -3\sqrt{6}\)

And \(-\sqrt{54} = -1 \times \sqrt{9 \times 6} = -3\sqrt{6 }

\)

We see that it is indeed the case. The two numbers are \(3\sqrt{3} \text{ and } -\sqrt{2}\)

Therefore,\( x = -3\sqrt{3} \text{ and } \sqrt{2}\)

The answers are Choices C and D

_________________

gmatclubot

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