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# What is the value of 9x^2 - 30xy + 25y^2 ?

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What is the value of 9x^2 - 30xy + 25y^2 ?  [#permalink]

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01 Mar 2016, 00:19
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What is the value of $$9x^2 - 30xy + 25y^2$$?

1) $$3x - 5y = 20$$

2) $$x + y = 12$$

___________________________________
I need to know how you derive 3x - 5y from the question stem.
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What is the value of 9x^2 - 30xy + 25y^2 ?  [#permalink]

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Updated on: 01 Mar 2016, 04:13
Nez wrote:
What is the value of $$9x^2 - 30xy + 25y^2$$?

1) $$3x - 5y = 20$$

2) $$x + y = 12$$

___________________________________
I need to know how you derive 3x - 5y from the question stem.

When you see the equation in the form of squares and terms of x and y, try to convert it in the from of $$a^2 +/- 2ab + b^2$$
Try to break the xy term in the from 2ab

In this case, observe that $$30xy = 2*3x*5y$$,
Now, see that $$9x^2 = (3x)^2$$ and $$25y^2 = (5y)^2$$

From here on, we can say that $$9x^2 - 30xy + 25y^2=(3x - 5y)^2$$

Reiterating: The key is to convert the xy term in the form 2ab.
Does this help?

Originally posted by TeamGMATIFY on 01 Mar 2016, 02:45.
Last edited by TeamGMATIFY on 01 Mar 2016, 04:13, edited 1 time in total.
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Posts: 7200
What is the value of 9x^2 - 30xy + 25y^2 ?  [#permalink]

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01 Mar 2016, 03:10
1
Nez wrote:
What is the value of $$9x^2 - 30xy + 25y^2$$?

1) $$3x - 5y = 20$$

2) $$x + y = 12$$

___________________________________
I need to know how you derive 3x - 5y from the question stem.

hi,

two ways..
1) you have three term one in x^2, one in xy and third in y^2..

also the coefficients of x^2 and y^2 are PERFECT square 9 and 25..

so $$(3x)^2- 30xy+(5y)^2$$..

now $$(a-b)^2=a^2-2ab+b^2$$... where $$a=3x$$ and $$b=5y$$..

so $$9x^2 - 30xy + 25y^2=(3x-5y)^2$$..

2) $$(3x)^2- 30xy+(5y)^2$$.

=> $$(3x)^2- \frac{30xy}{2} - \frac{30xy}{2} +(5y)^2$$..

=> $$(3x)^2- 15xy-15xy+(5y)^2$$..

=> $$(3x)^2- 3*5*x*y-3*5*x*y+(5y)^2$$..

$$3x(3x-5y)-5y(3x-5y)= (3x-5y)(3x-5y)$$

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Re: What is the value of 9x^2 - 30xy + 25y^2 ?  [#permalink]

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01 Mar 2016, 04:02
What's the point in your solution, if i dont see what you solved?
I can tell better what it is if you encode them properly. Thanks.
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Posts: 52149
Re: What is the value of 9x^2 - 30xy + 25y^2 ?  [#permalink]

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01 Mar 2016, 04:08
1
Nez wrote:
What's the point in your solution, if i dont see what you solved?
I can tell better what it is if you encode them properly. Thanks.

Edited the posts above.

Hope it helps.
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Re: What is the value of 9x^2 - 30xy + 25y^2 ?  [#permalink]

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01 Mar 2016, 04:16
1
Nez wrote:
What's the point in your solution, if i dont see what you solved?
I can tell better what it is if you encode them properly. Thanks.

Hi Nez,

Sorry for the post.
Hope it helps now.
Bunuel Thanks for the editing.
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Re: What is the value of 9x^2 - 30xy + 25y^2 ?  [#permalink]

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01 Mar 2016, 13:16
chetan2u wrote:
Nez wrote:
What is the value of $$9x^2 - 30xy + 25y^2$$?

1) $$3x - 5y = 20$$

2) $$x + y = 12$$

___________________________________
I need to know how you derive 3x - 5y from the question stem.

hi,

two ways..
1) you have three term one in x^2, one in xy and third in y^2..

also the coefficients of x^2 and y^2 are PERFECT square 9 and 25..

so $$(3x)^2- 30xy+(5y)^2$$..

now $$(a-b)^2=a^2-2ab+b^2$$... where $$a=3x$$ and $$b=5y$$..

so $$9x^2 - 30xy + 25y^2=(3x-5y)^2$$..

2) $$(3x)^2- 30xy+(5y)^2$$.

=> $$(3x)^2- \frac{30xy}{2} - \frac{30xy}{2} +(5y)^2$$..

=> $$(3x)^2- 15xy-15xy+(5y)^2$$..

=> $$(3x)^2- 3*5*x*y-3*5*x*y+(5y)^2$$..

$$3x(3x-5y)-5y(3x-5y)= (3x-5y)(3x-5y)$$

This is beauty
$$(3x)^2- \frac{30xy}{2} - \frac{30xy}{2} +(5y)^2$$..

=> $$(3x)^2- 15xy-15xy+(5y)^2$$..

=> $$(3x)^2- 3*5*x*y-3*5*x*y+(5y)^2$$..

$$3x(3x-5y)-5y(3x-5y)= (3x-5y)(3x-5y)$$

didn't know it would not just be only easy but beautiful as well
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What is the value of 9x^2 - 30xy + 25y^2 ?  [#permalink]

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01 Mar 2016, 14:13
1
Nez wrote:
chetan2u wrote:
Nez wrote:
What is the value of $$9x^2 - 30xy + 25y^2$$?

1) $$3x - 5y = 20$$

2) $$x + y = 12$$

___________________________________
I need to know how you derive 3x - 5y from the question stem.

hi,

two ways..
1) you have three term one in x^2, one in xy and third in y^2..

also the coefficients of x^2 and y^2 are PERFECT square 9 and 25..

so $$(3x)^2- 30xy+(5y)^2$$..

now $$(a-b)^2=a^2-2ab+b^2$$... where $$a=3x$$ and $$b=5y$$..

so $$9x^2 - 30xy + 25y^2=(3x-5y)^2$$..

2) $$(3x)^2- 30xy+(5y)^2$$.

=> $$(3x)^2- \frac{30xy}{2} - \frac{30xy}{2} +(5y)^2$$..

=> $$(3x)^2- 15xy-15xy+(5y)^2$$..

=> $$(3x)^2- 3*5*x*y-3*5*x*y+(5y)^2$$..

$$3x(3x-5y)-5y(3x-5y)= (3x-5y)(3x-5y)$$

This is beauty
$$(3x)^2- \frac{30xy}{2} - \frac{30xy}{2} +(5y)^2$$..

=> $$(3x)^2- 15xy-15xy+(5y)^2$$..

=> $$(3x)^2- 3*5*x*y-3*5*x*y+(5y)^2$$..

$$3x(3x-5y)-5y(3x-5y)= (3x-5y)(3x-5y)$$

y^2
didn't know it would not just be only easy but beautiful as well

There is a formulaic approach to solving quadratic equations as shown below:

For the quadratic equation, ax^2+bx+c=0, with x being the variable and a,b,c are constants,

Roots $$x_1$$ and $$x_2$$$$= \frac {-b \pm \sqrt{b^2-4ac}}{2*a}$$ and then you can write the quadratic equation as : $$(x-x_1)(x-x_2)=0$$

Thus for the question above,

$$9x^2−30xy+25y^2$$---> $$a=9, b=-30y, c=25y^2$$, you get,

Thus, $$x = \frac {-b \pm \sqrt{b^2-4ac}}{2*a} = \frac {30y \pm \sqrt{(30y)^2-4*9*25y^2}}{2*9} = \frac {30y \pm \sqrt{900*y^2-900*y^2}}{2*9} = \frac {30y}{2*9} = \frac {5y}{3}$$

---> You get $$(x-\frac{5*y}{3})(x-\frac{5*y}{3})=0$$ as the equation ---> $$(3x-5y)^2 = 0$$ as the perfect square.

Again, this method might seem a bit excessive for time but for some questions where factorization isnt striaghtforward, you can use this to find the roots of the quadratic equation. Alternately, in order to check whether a given qudratic equation is a perfect square is to check for the discriminant.

For a given perfect square, Discriminant, $$D= \sqrt{b^2-4ac}= 0$$. In the case above as well you got D = 0, thus telling you that the given quadratic equation is a perfect square.

Hope this helps.
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Re: What is the value of 9x^2 - 30xy + 25y^2 ?  [#permalink]

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02 Mar 2016, 11:40
Quote:
There is a formulaic approach to solving quadratic equations as shown below:

For the quadratic equation, ax^2+bx+c=0, with x being the variable and a,b,c are constants,

Roots $$x_1$$ and $$x_2$$$$= \frac {-b \pm \sqrt{b^2-4ac}}{2*a}$$ and then you can write the quadratic equation as : $$(x-x_1)(x-x_2)=0$$

Thus for the question above,

$$9x^2−30xy+25y^2$$---> $$a=9, b=-30y, c=25y^2$$, you get,

Thus, $$x = \frac {-b \pm \sqrt{b^2-4ac}}{2*a} = \frac {30y \pm \sqrt{(30y)^2-4*9*25y^2}}{2*9} = \frac {30y \pm \sqrt{900*y^2-900*y^2}}{2*9} = \frac {30y}{2*9} = \frac {5y}{3}$$

---> You get $$(x-\frac{5*y}{3})(x-\frac{5*y}{3})=0$$ as the equation ---> $$(3x-5y)^2 = 0$$ as the perfect square.

Again, this method might seem a bit excessive for time but for some questions where factorization isnt striaghtforward, you can use this to find the roots of the quadratic equation. Alternately, in order to check whether a given qudratic equation is a perfect square is to check for the discriminant.

For a given perfect square, Discriminant, $$D= \sqrt{b^2-4ac}= 0$$. In the case above as well you got D = 0, thus telling you that the given quadratic equation is a perfect square.

Hope this helps.

I'm yet to come across a GMAT question requiring the quadratic formula.
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Re: What is the value of 9x^2 - 30xy + 25y^2 ?  [#permalink]

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02 Mar 2016, 12:03
Nez wrote:
Quote:
There is a formulaic approach to solving quadratic equations as shown below:

For the quadratic equation, ax^2+bx+c=0, with x being the variable and a,b,c are constants,

Roots $$x_1$$ and $$x_2$$$$= \frac {-b \pm \sqrt{b^2-4ac}}{2*a}$$ and then you can write the quadratic equation as : $$(x-x_1)(x-x_2)=0$$

Thus for the question above,

$$9x^2−30xy+25y^2$$---> $$a=9, b=-30y, c=25y^2$$, you get,

Thus, $$x = \frac {-b \pm \sqrt{b^2-4ac}}{2*a} = \frac {30y \pm \sqrt{(30y)^2-4*9*25y^2}}{2*9} = \frac {30y \pm \sqrt{900*y^2-900*y^2}}{2*9} = \frac {30y}{2*9} = \frac {5y}{3}$$

---> You get $$(x-\frac{5*y}{3})(x-\frac{5*y}{3})=0$$ as the equation ---> $$(3x-5y)^2 = 0$$ as the perfect square.

Again, this method might seem a bit excessive for time but for some questions where factorization isnt striaghtforward, you can use this to find the roots of the quadratic equation. Alternately, in order to check whether a given qudratic equation is a perfect square is to check for the discriminant.

For a given perfect square, Discriminant, $$D= \sqrt{b^2-4ac}= 0$$. In the case above as well you got D = 0, thus telling you that the given quadratic equation is a perfect square.

Hope this helps.

I'm yet to come across a GMAT question requiring the quadratic formula.

I mentioned the formula to give you the complete picture. Additionally, you can check whether a given quadratic equation is in fact a perfect square and not spend time on trying to coming up with a perfect square expression for a quadratic equation that is in fact not a perfect square.
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Re: What is the value of 9x^2 - 30xy + 25y^2 ?  [#permalink]

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02 Mar 2016, 22:05
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

What is the value of 9x^2−30xy+25y^2 ?

1) 3x−5y=20

2) x+y=12

When you modify the original condition and the question, they become 9x^2 - 30xy + 25y^2=(3x-5y)^2?. Since 1) is unique, the answer is A.

 Once we modify the original condition and the question according to the variable approach method 1, we can solve approximately 30% of DS questions.
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Re: What is the value of 9x^2 - 30xy + 25y^2 ?  [#permalink]

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02 Nov 2018, 16:26
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