Form an equation $a x^{2} + b x y + c y^{2}$ by subtracting the sum of $x^{2} - 5 x y + 2 y^{2}$ and $y^{2} - 2 x y - 3 x^{2}$ from the sum of $6 x^{2} - 8 x y - y^{2}$ and $2 x y - 2 y^{2} - x^{2}$ . Find $a + b + c$ .

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Solution

We first add the expressions $x^{2} - 5 x y + 2 y^{2}$ and $y^{2} - 2 x y - 3 x^{2}$ . And then, we add $6 x^{2} - 8 x y - y^{2}$ and $2 x y - 2 y^{2} - x^{2}$ as shown below:

$(x^{2} - 5 x y + 2 y^{2}) + (y^{2} - 2 x y - 3 x^{2}) = x^{2} - 5 x y + 2 y^{2} + y^{2} - 2 x y - 3 x^{2} = (x^{2} - 3 x^{2}) + (2 y^{2} + y^{2}) + (- 5 x y - 2 x y) (C o m b i n i n g l i k e t e r m s) = - 2 x^{2} + 3 y^{2} - 7 x y . . . . (1)$

$(6 x^{2} - 8 x y - y^{2}) + (2 x y - 2 y^{2} - x^{2}) = 6 x^{2} - 8 x y - y^{2} + 2 x y - 2 y^{2} - x^{2} = (6 x^{2} - x^{2}) + (- y^{2} - 2 y^{2}) + (- 8 x y + 2 x y) (C o m b i n i n g l i k e t e r m s) = 5 x^{2} - 3 y^{2} - 6 x y . . . . (2)$

Now, as per the question, we subtract equation 1 from equation 2 and equate it to $a x^{2} + b x y + c y^{2}$ as follows:

$(5 x^{2} - 3 y^{2} - 6 x y) - (- 2 x^{2} + 3 y^{2} - 7 x y) = a x^{2} + b x y + c y^{2} = 5 x^{2} - 3 y^{2} - 6 x y + 2 x^{2} - 3 y^{2} + 7 x y = a x^{2} + b x y + c y^{2} = (5 x^{2} + 2 x^{2}) + (- 3 y^{2} - 3 y^{2}) + (- 6 x y + 7 x y) = a x^{2} + b x y + c y^{2} (C o m b i n i n g l i k e t e r m s) = 7 x^{2} - 6 y^{2} + x y = a x^{2} + b x y + c y^{2} \Rightarrow a = 7, b = 1, c = - 6 (B y c o m p a r i n g c o e f f i c i e n t s o f x^{2}, y^{2} a n d x y)$

Now, adding $a + b + c$ , we get:

$a + b + c = 7 + 1 - 6 = 8 - 6 = 2$

Hence, $a + b + c = 2$ .

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Form an equation ax2+bxy+cy2 by subtracting the sum of x2−5xy+2y2 and y2−2xy−3x2 from the sum of 6x2−8xy−y2 and 2xy−2y2−x2. Find a+b+c.

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