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Vlad [161]
3 years ago
7

7. Fred had 85 marbles. After a few games, he had 120% of his original number. How many

Mathematics
1 answer:
Gnoma [55]3 years ago
4 0

Answer:

102 marbles

Step-by-step explanation:

100% of 85 is 85.

20% of 85 = 17

85 + 17 = 102

Hope that helps!

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You need to complete the square in x and in y,

x^{2}  + y^{2} + 6x + 4y + 10 = 1

First, group the x-terms and the y-terms separately.

x^{2} + 6x  + y^{2} + 4y + 10 = 1

Move the 10 to the right side by subtracting 10 from both sides.

x^{2} + 6x  + y^{2} + 4y = -9

Now complete the square in x and in y.
The constant you need to add to complete each square is the square of half of the coefficient of the x or y term. Make sure to add the constants to both sides of the equation.

x^{2} + 6x  + 9 + y^{2} + 4y + 4= -9 + 9 + 4

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Resolve the expression <img src="https://tex.z-dn.net/?f=%5Cfrac%7B%28x-2%29%7D%7B%28x%5E%7B2%7D%20%2B1%29%28x-1%29%5E%7B2%7D%20
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The partial fraction expansion will look like

(<em>x</em> - 2)/((<em>x</em> ² + 1) (<em>x</em> - 1)²) = (<em>ax</em> + <em>b</em>)/(<em>x</em> ² + 1) + <em>c</em>/(<em>x</em> - 1) + <em>d</em>/(<em>x</em> - 1)²

Get everything in terms of a common denominator, and compare the numerators on both sides:

<em>x</em> - 2 = (<em>ax</em> + <em>b</em>) (<em>x</em> - 1)² + <em>c</em> (<em>x</em> ² + 1) (<em>x</em> - 1) + <em>d</em> (<em>x</em> ² + 1)

Expand the right side:

<em>x</em> - 2 = (<em>ax</em> + <em>b</em>) (<em>x</em> - 1)² + <em>c</em> (<em>x</em> ² + 1) (<em>x</em> - 1) + <em>d</em> (<em>x</em> ² + 1)

<em>x</em> - 2 = (<em>a</em> + <em>c</em>) <em>x</em> ³ + (-2<em>a</em> + <em>b</em> - <em>c</em> + <em>d</em>) <em>x</em> ² + (<em>a</em> - 2<em>b</em> + <em>c</em>) <em>x</em> + <em>b</em> - <em>c</em> + <em>d</em>

<em />

Match up the coefficients and solve the resulting system of equations:

<em>a</em> + <em>c</em> = 0

-2<em>a</em> + <em>b</em> - <em>c</em> + <em>d</em> = 0

<em>a</em> - 2<em>b</em> + <em>c</em> = 1

<em>b</em> - <em>c</em> + <em>d</em> = -2

==>   <em>a</em> = -1, <em>b</em> = -1/2, <em>c</em> = 1, <em>d</em> = -1/2

So the expansion into partial fractions is

(<em>x</em> - 2)/((<em>x</em> ² + 1) (<em>x</em> - 1)²) = (-<em>x</em> - 1/2)/(<em>x</em> ² + 1) + 1/(<em>x</em> - 1) - 1/(2 (<em>x</em> - 1)²)

… = -(2<em>x</em> + 1)/(2 (<em>x</em> ² + 1)) + 1/(<em>x</em> - 1) - 1/(2 (<em>x</em> - 1)²)

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