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Math Expert V
Joined: 02 Sep 2009
Posts: 55272
Re: If root{3-2x} = root(2x) +1, then 4x^2 =  [#permalink]

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reto wrote:
Bunuel wrote:
If $$\sqrt{3-2x} = \sqrt{2x} +1$$, then $$4x^2$$ =

(A) 1
(B) 4
(C) 2 − 2x
(D) 4x − 2
(E) 6x − 1

Diagnostic Test
Question: 16
Page: 22
Difficulty: 600

Why is this question categorized as difficulty 700 above and 600 in the question steam? What's the actual difficulty?

If you square the expression $$\sqrt{2x} +1$$ and there are NO paranthesis, why is it not correct to square each term seperately to get 2x + 1? Thank you so much

1. 600 difficulty was given when posted. 700 difficulty is based on users' answers on the question.
2. (a + b)^2 generally does not equal to a^2 + b^2. I'ts pretty basic actually.
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Re: If root{3-2x} = root(2x) +1, then 4x^2 =  [#permalink]

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Bunuel wrote:
reto wrote:
Bunuel wrote:
If $$\sqrt{3-2x} = \sqrt{2x} +1$$, then $$4x^2$$ =

(A) 1
(B) 4
(C) 2 − 2x
(D) 4x − 2
(E) 6x − 1

Diagnostic Test
Question: 16
Page: 22
Difficulty: 600

Why is this question categorized as difficulty 700 above and 600 in the question steam? What's the actual difficulty?

If you square the expression $$\sqrt{2x} +1$$ and there are NO paranthesis, why is it not correct to square each term seperately to get 2x + 1? Thank you so much

1. 600 difficulty was given when posted. 700 difficulty is based on users' answers on the question.
2. (a + b)^2 generally does not equal to a^2 + b^2. I'ts pretty basic actually.

Thanks. Thats what I meant in the question there are no paranthesis, but when you squared it in your solution, you gave the expression extra paranthesis - why? I know it's not the same. But I don't understand why you completed it when squaring...
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Math Expert V
Joined: 02 Sep 2009
Posts: 55272
Re: If root{3-2x} = root(2x) +1, then 4x^2 =  [#permalink]

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1
reto wrote:
Thanks. Thats what I meant in the question there are no paranthesis, but when you squared it in your solution, you gave the expression extra paranthesis - why? I know it's not the same. But I don't understand why you completed it when squaring...

Squaring means multiplying by the same number, so when you square a + b, you get (a + b)(a + b) = (a + b)^2.

I'd advice to brush up fundamentals before practicing questions, especially, hard ones.
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Re: If root{3-2x} = root(2x) +1, then 4x^2 =  [#permalink]

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1
Bunuel wrote:
If $$\sqrt{3-2x} = \sqrt{2x} +1$$, then $$4x^2$$ =

(A) 1
(B) 4
(C) 2 − 2x
(D) 4x − 2
(E) 6x − 1

Diagnostic Test
Question: 16
Page: 22
Difficulty: 600

sqrt(3-2x)-1 = sqrt(2x) | square everything
2x = 3-2x - 2*sqrt(3-2x)+1
2x=4 - 2x - 2*sqrt(3-2x)
4-4x=2*sqrt(3-2x) | square everything again
16+16x^2 -32x = 12 - 8x | add 8x to both sides, subtract 12 from both sides
16x^2 - 24x +4 = 0 | divide by 4
4x^2 -6x +1 = 0 | add 6x to both sides and subtract 1:
4x^2 = 6x-1

E
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Re: If root{3-2x} = root(2x) +1, then 4x^2 =  [#permalink]

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Bunuel wrote:
If $$\sqrt{3-2x} = \sqrt{2x} +1$$, then $$4x^2$$ =

(A) 1
(B) 4
(C) 2 − 2x
(D) 4x − 2
(E) 6x − 1

Diagnostic Test
Question: 16
Page: 22
Difficulty: 600

Main Idea:While simplifying keep the more complex root expression alone on one side and simplify.

Details: sqrt(3-2x)=sqrt(2) +1.

Squaring on both sides: 3-2x=2x+1+2*sqrt(2x)

Simplifying we get (4x)^2 = 6x-1

Hence E.
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Re: If root{3-2x} = root(2x) +1, then 4x^2 =  [#permalink]

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Bunuel wrote:
SOLUTION

If $$\sqrt{3-2x} = \sqrt{2x} +1$$, then $$4x^2$$ =

(A) 1
(B) 4
(C) 2 − 2x
(D) 4x − 2
(E) 6x − 1

$$\sqrt{3-2x} = \sqrt{2x} +1$$ --> square both sides: $$(\sqrt{3-2x})^2 =(\sqrt{2x} +1)^2$$ --> $$3-2x=2x+2*\sqrt{2x}+1$$ --> rearrange so that to have root at one side: $$2-4x=2*\sqrt{2x}$$ --> reduce by 2: $$1-2x=\sqrt{2x}$$ --> square again: $$(1-2x)^2=(\sqrt{2x})^2$$ --> $$1-4x+4x^2=2x$$ --> rearrange again: $$4x^2=6x-1$$.

Answer: E.

Bunuel can you please share your expertise knowledge on the following how from this $$(\sqrt{3-2x})^2 =(\sqrt{2x} +1)^2$$ we get this $$3-2x=2x+2*\sqrt{2x}+1$$ ? thank you for your kind attention to my question Math Expert V
Joined: 02 Sep 2009
Posts: 55272
If root{3-2x} = root(2x) +1, then 4x^2 =  [#permalink]

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dave13 wrote:
Bunuel wrote:
SOLUTION

If $$\sqrt{3-2x} = \sqrt{2x} +1$$, then $$4x^2$$ =

(A) 1
(B) 4
(C) 2 − 2x
(D) 4x − 2
(E) 6x − 1

$$\sqrt{3-2x} = \sqrt{2x} +1$$ --> square both sides: $$(\sqrt{3-2x})^2 =(\sqrt{2x} +1)^2$$ --> $$3-2x=2x+2*\sqrt{2x}+1$$ --> rearrange so that to have root at one side: $$2-4x=2*\sqrt{2x}$$ --> reduce by 2: $$1-2x=\sqrt{2x}$$ --> square again: $$(1-2x)^2=(\sqrt{2x})^2$$ --> $$1-4x+4x^2=2x$$ --> rearrange again: $$4x^2=6x-1$$.

Answer: E.

Bunuel can you please share your expertise knowledge on the following how from this $$(\sqrt{3-2x})^2 =(\sqrt{2x} +1)^2$$ we get this $$3-2x=2x+2*\sqrt{2x}+1$$ ? thank you for your kind attention to my question Let me asks you:

1. What does $$(\sqrt{a})^2$$ equal to?
2. What does $$(a + b)^2$$ equal to?

If you can answer these questions, you should be able to figure out your doubt.
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VP  D
Joined: 09 Mar 2016
Posts: 1283
If root{3-2x} = root(2x) +1, then 4x^2 =  [#permalink]

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Bunuel wrote:
dave13 wrote:
Bunuel wrote:
SOLUTION

If $$\sqrt{3-2x} = \sqrt{2x} +1$$, then $$4x^2$$ =

(A) 1
(B) 4
(C) 2 − 2x
(D) 4x − 2
(E) 6x − 1

$$\sqrt{3-2x} = \sqrt{2x} +1$$ --> square both sides: $$(\sqrt{3-2x})^2 =(\sqrt{2x} +1)^2$$ --> $$3-2x=2x+2*\sqrt{2x}+1$$ --> rearrange so that to have root at one side: $$2-4x=2*\sqrt{2x}$$ --> reduce by 2: $$1-2x=\sqrt{2x}$$ --> square again: $$(1-2x)^2=(\sqrt{2x})^2$$ --> $$1-4x+4x^2=2x$$ --> rearrange again: $$4x^2=6x-1$$.

Answer: E.

Bunuel can you please share your expertise knowledge on the following how from this $$(\sqrt{3-2x})^2 =(\sqrt{2x} +1)^2$$ we get this $$3-2x=2x+2*\sqrt{2x}+1$$ ? thank you for your kind attention to my question Let me asks you:

1. What does $$(\sqrt{a})^2$$ equal to?
2. What does $$(a + b)^2$$ equal to?

If you can answer these questions, you should be able to figure out your doubt.

Hi Bunuel

thanks for your questions, i will gladly answer your questions:)

1. What does $$(\sqrt{a})^2$$ equal to? it equals $$a$$

2. What does $$(a + b)^2$$ equal to? it equals this $$(a+b)(a+b) = a^2+ab+ba+b^2 = a^2+2ab+b^2$$

now let me get back to the question

$$(\sqrt{3-2x})^2 = (\sqrt{2x} +1)^2$$ square both sides (using this formula $$(\sqrt{a})^2$$ = $$a$$

so we get $$(3-2x)^2 = (2x-1)^2$$

now using this formula $$(a + b)^2$$ = $$a^2+2ab+b^2$$

we get $$(3-2x) ( 3-2x) = (2x+1) (2x+1)$$ so from here we get $$10=16x$$:?

can you please explain what am i doing wrong knowing formula is one thing, but applying it correctly is quite challenging for me ......sometimes Math Expert V
Joined: 02 Sep 2009
Posts: 55272
Re: If root{3-2x} = root(2x) +1, then 4x^2 =  [#permalink]

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1
dave13 wrote:
Bunuel wrote:
dave13 wrote:
If $$\sqrt{3-2x} = \sqrt{2x} +1$$, then $$4x^2$$ =

(A) 1
(B) 4
(C) 2 − 2x
(D) 4x − 2
(E) 6x − 1

$$\sqrt{3-2x} = \sqrt{2x} +1$$ --> square both sides: $$(\sqrt{3-2x})^2 =(\sqrt{2x} +1)^2$$ --> $$3-2x=2x+2*\sqrt{2x}+1$$ --> rearrange so that to have root at one side: $$2-4x=2*\sqrt{2x}$$ --> reduce by 2: $$1-2x=\sqrt{2x}$$ --> square again: $$(1-2x)^2=(\sqrt{2x})^2$$ --> $$1-4x+4x^2=2x$$ --> rearrange again: $$4x^2=6x-1$$.

Answer: E.

Bunuel can you please share your expertise knowledge on the following how from this $$(\sqrt{3-2x})^2 =(\sqrt{2x} +1)^2$$ we get this $$3-2x=2x+2*\sqrt{2x}+1$$ ? thank you for your kind attention to my question Let me asks you:

1. What does $$(\sqrt{a})^2$$ equal to?
2. What does $$(a + b)^2$$ equal to?

If you can answer these questions, you should be able to figure out your doubt.

Hi Bunuel

thanks for your questions, i will gladly answer your questions:)

1. What does $$(\sqrt{a})^2$$ equal to? it equals $$a$$

2. What does $$(a + b)^2$$ equal to? it equals this $$(a+b)(a+b) = a^2+ab+ba+b^2 = a^2+2ab+b^2$$

now let me get back to the question

$$(\sqrt{3-2x})^2 = (\sqrt{2x} +1)^2$$ square both sides (using this formula $$(\sqrt{a})^2$$ = $$a$$

so we get $$(3-2x)^2 = (2x-1)^2$$

now using this formula $$(a + b)^2$$ = $$a^2+2ab+b^2$$

we get $$(3-2x) ( 3-2x) = (2x+1) (2x+1)$$ so from here we get $$10=16x$$:?

can you please explain what am i doing wrong knowing formula is one thing, but applying it correctly is quite challenging for me ......sometimes 1. Yes, $$(\sqrt{a})^2=a$$, hence $$(\sqrt{3-2x})^2 =3-2x$$.

2. Here you are also correct: $$(a + b)^2=a^2+2ab+b^2$$, hence $$(\sqrt{2x} +1)^2=(\sqrt{2x})^2 + 2*\sqrt{2x}*1+1^2=2x+2*\sqrt{2x}+1$$.

Does this make sense?
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If root{3-2x} = root(2x) +1, then 4x^2 =  [#permalink]

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Bunuel, it makes perfect sense ! i applied this formula $$(a - b)^2$$ = $$a^2-2ab+b^2$$ to $$(1-2x)^2=(\sqrt{2x})^2$$

so i get this $$1^2 - 2 * 1 *(-2x) +2x^2 = 2x$$ ----> $$1 -2-2x+4x=2x$$ --> $$1 -2+2x= 2x$$ ---> $$-1 =0$$:? what am i doing wrong , can you please explain in two or three words many thanks! Math Expert V
Joined: 02 Sep 2009
Posts: 55272
Re: If root{3-2x} = root(2x) +1, then 4x^2 =  [#permalink]

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1
dave13 wrote:
Bunuel, it makes perfect sense ! i applied this formula $$(a - b)^2$$ = $$a^2-2ab+b^2$$ to $$(1-2x)^2=(\sqrt{2x})^2$$

so i get this 1^2 - 2 * 1 *(-2x) +2x^2 = 2x ----> $$1 -2-2x+4x=2x$$ --> $$1 -2+2x= 2x$$ ---> $$-1 =0$$:? what am i doing wrong , can you please explain in two or three words many thanks! I highlighted the mistakes.

$$(1 - 2x)^2 = 1 - 2*2x + (2x)^2 = 1 - 4x + 4x^2$$
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If root{3-2x} = root(2x) +1, then 4x^2 =  [#permalink]

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Bunuel wrote:
dave13 wrote:
Bunuel, it makes perfect sense ! i applied this formula $$(a - b)^2$$ = $$a^2-2ab+b^2$$ to $$(1-2x)^2=(\sqrt{2x})^2$$

so i get this 1^2 - 2 * 1 *(-2x) +2x^2 = 2x ----> $$1 -2-2x+4x=2x$$ --> $$1 -2+2x= 2x$$ ---> $$-1 =0$$:? what am i doing wrong , can you please explain in two or three words many thanks! I highlighted the mistakes.

$$(1 - 2x)^2 = 1 - 2*2x + (2x)^2 = 1 - 4x + 4x^2$$

Bunuel thank you ! just last question how did you convert this $$(2x)^2$$ and got $$4x^2$$ $$(2x)^2$$ equals $$4x$$ and not $$4x^2$$

please explain this part and I will be more than happy Math Expert V
Joined: 02 Sep 2009
Posts: 55272
Re: If root{3-2x} = root(2x) +1, then 4x^2 =  [#permalink]

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1
dave13 wrote:
Bunuel wrote:
dave13 wrote:
Bunuel, it makes perfect sense ! i applied this formula $$(a - b)^2$$ = $$a^2-2ab+b^2$$ to $$(1-2x)^2=(\sqrt{2x})^2$$

so i get this 1^2 - 2 * 1 *(-2x) +2x^2 = 2x ----> $$1 -2-2x+4x=2x$$ --> $$1 -2+2x= 2x$$ ---> $$-1 =0$$:? what am i doing wrong , can you please explain in two or three words many thanks! I highlighted the mistakes.

$$(1 - 2x)^2 = 1 - 2*2x + (2x)^2 = 1 - 4x + 4x^2$$

Bunuel thank you ! just last question how did you convert this $$(2x)^2$$ and got $$4x^2$$ $$(2x)^2$$ equals $$4x$$and not $$4x^2$$

please explain this part and I will be more than happy We have $$(2x)^2$$. Notice that we are squaring 2x, so you should square both 2 and x: $$(2x)^2=2x*2x=4x^2$$
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Re: If root{3-2x} = root(2x) +1, then 4x^2 =  [#permalink]

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this is tricky
the key here is to arrange the equation so that we have root at one side. when this happen, we can square and after squaring, we can have simple result.
we square the first time immediately because the left side is a root already
after getting the result fo the first squaring, we arrange so that we have root at one side.

look at the way Bunnuel do. I just explain more.
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Re: If root{3-2x} = root(2x) +1, then 4x^2 =  [#permalink]

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1
Bunuel wrote:
If $$\sqrt{3-2x} = \sqrt{2x} +1$$, then $$4x^2$$ =

(A) 1
(B) 4
(C) 2 − 2x
(D) 4x − 2
(E) 6x − 1

Diagnostic Test
Question: 16
Page: 22
Difficulty: 600

$$\sqrt{3-2x} = \sqrt{2x} +1$$

Square once to get
3-2x = 2x + 1 + 2$$\sqrt{2x}$$
1-2x = $$\sqrt{2x}$$

Square again to get
$$(1-2x)^2$$ = 2x
4x^2 = 6x - 1

E
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Re: If root{3-2x} = root(2x) +1, then 4x^2 =  [#permalink]

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Top Contributor
Bunuel wrote:
If $$\sqrt{3-2x} = \sqrt{2x} +1$$, then $$4x^2$$ =

(A) 1
(B) 4
(C) 2 − 2x
(D) 4x − 2
(E) 6x − 1

GIVEN: $$\sqrt{3-2x} = \sqrt{2x} +1$$

Square both sides to get: $$(\sqrt{3-2x})^2 =(\sqrt{2x} +1)^2$$

Expand and simplify to get: $$3-2x=2x+2\sqrt{2x}+1$$

Subtract 1 from both sides to get: $$1-2x=2x+2\sqrt{2x}$$

Subtract 2x from both sides to get: $$2-4x=2\sqrt{2x}$$

Divide both sides by 2 to get: $$1-2x=\sqrt{2x}$$

Square both sides to get: $$(1-2x)^2=(\sqrt{2x})^2$$

Expand and simplify to get: $$1-4x+4x^2=2x$$

Add 4x to both sides: $$1 + 4x^2=6x$$

Subtract 1 from both sides to get: $$4x^2=6x-1$$

Answer: E

Cheers,
Brent
_________________ Re: If root{3-2x} = root(2x) +1, then 4x^2 =   [#permalink] 18 Mar 2019, 10:07

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