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2 years ago

Please help giving brainiest

Mathematics

1 answer:

ser-zykov [4K]2 years ago

3 0

Answer: 1,000 units, 1300 dollars

Step-by-step explanation:

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Write out the form of the partial fraction decomposition of the function. Do not determine the numerical values of the coefficie

Dvinal [7]

For part (a), you have

$\dfrac x{x^2+x-6}=\dfrac x{(x+3)(x-2)}=\dfrac a{x+3}+\dfrac b{x-2}$
$x=a(x-2)+b(x+3)$

If $x=2$ , then $2=b(2-3)\implies b=-2$ .

If $x=-3$ , then $-3=a(-3-2)\implies a=\dfrac35$ .

So,

$\dfrac x{x^2+x-6}=\dfrac 3{5(x+3)}-\dfrac 2{x-2}$

For part (b), since the degrees of the numerator and denominator are the same, you first need to find the quotient and remainder upon division.

$\dfrac{x^2}{x^2+x+2}=\dfrac{x^2+x+2-x-2}{x^2+x+2}=1-\dfrac{x+2}{x^2+x+2}$

In the remainder term, the denominator $x^2+x+2$ can't be factorized into linear components with real coefficients, since the discriminant is negative $(1-4\times1\times2=-7)$ . However, you can still factorized over the complex numbers, so a partial fraction decomposition in terms of complexes does exist.

$x^2+x+2=0\implies x=-\dfrac12\pm\dfrac{\sqrt7}2i$
$\implies x^2+x+2=\left(x-\left(-\dfrac12+\dfrac{\sqrt7}2i\right)\right)\left(x-\left(-\dfrac12-\dfrac{\sqrt7}2i\right)\right)$
$\implies x^2+x+2=\left(x+\dfrac12-\dfrac{\sqrt7}2i\right)\left(x+\dfrac12+\dfrac{\sqrt7}2i\right)$

Then you have

$\dfrac{x+2}{x^2+x+2}=\dfrac a{x+\dfrac12-\dfrac{\sqrt7}2i}+\dfrac b{x+\dfrac12+\dfrac{\sqrt7}2i}$
$x+2=a\left(x+\dfrac12+\dfrac{\sqrt7}2i\right)+b\left(x+\dfrac12-\dfrac{\sqrt7}2i\right)$

When $x=-\dfrac12-\dfrac{\sqrt7}2i$ , you have

$-\dfrac12-\dfrac{\sqrt7}2i+2=b\left(-\dfrac12-\dfrac{\sqrt7}2i+\dfrac12-\dfrac{\sqrt7}2i\right)$
$\dfrac32-\dfrac{\sqrt7}2i=-\sqrt7ib$
$b=\dfrac12+\dfrac3{2\sqrt7}i=\dfrac1{14}(7+3\sqrt7i)$

When $x=-\dfrac12+\dfrac{\sqrt7}2i$ , you have

$-\dfrac12+\dfrac{\sqrt7}2i+2=a\left(-\dfrac12+\dfrac{\sqrt7}2i+\dfrac12+\dfrac{\sqrt7}2i\right)$
$\dfrac32+\dfrac{\sqrt7}2i=\sqrt7ia$
$a=\dfrac12-\dfrac3{2\sqrt7}i=\dfrac1{14}(7-3\sqrt7i)$

So, you could write

$\dfrac{x^2}{x^2+x+2}=1-\dfrac{x+2}{x^2+x+2}=1-\dfrac {7-3\sqrt7i}{14\left(x+\dfrac12-\dfrac{\sqrt7}2i\right)}-\dfrac {7+3\sqrt7i}{14\left(x+\dfrac12+\dfrac{\sqrt7}2i\right)}$

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5 0

3 years ago

Read 2 more answers

draw a basic market graph. Title it the market for the product in your news article. Label axes, curves, eqilibruim price and eq

Vaselesa [24]

The basic market graph is attached below.

<h3>What do you mean by market equilibrium?</h3>

A market is said to have reached equilibrium price when the supply of goods matches demand.

The equilibrium price is the only price where the plans of consumers and the plans of producers agree, where the amount consumers want to buy of the product, quantity demanded, is equal to the amount producers want to sell, quantity supplied. This common quantity is called the equilibrium quantity.

Quantity of gasoline:

Explanation of graph: The demand curve, D, and the supply curve, S, intersect at the equilibrium point E, with an equilibrium price of 1.4 dollars and an equilibrium quantity of 600.

The equilibrium is the only price where quantity demanded is equal to quantity supplied. At a price above equilibrium, like 1.8 dollars, quantity supplied exceeds the quantity demanded, so there is excess supply.

At a price below equilibrium, such as 1.2 dollars, quantity demanded exceeds quantity supplied, so there is excess demand.

Learn more about equilibrium quantity here:

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6 0

2 years ago

if you substract 1/2 from a number and multiply 1/2 from the result you get 1/8 . what is the number

qwelly [4]

3/4

Three fourths subtracted from one half is one fourth. One fourth times one half is one eighth.

5 0

3 years ago

Mario and his crew are headed to the beach. They start with 1500 coins and plan to spend 60 coins on one umbrella for the trip a

zysi [14]

Answer:

they have been on vacation for 2 days i think

Step-by-step explanation:

1500-60=1440

1440-1080= 360

360÷180=2

hope this helps

8 0

3 years ago

Hello pa help po pre cal yan

aliya0001 [1]

Answer:

I would love to help, Could you put the question in English?

Step-by-step explanation:

5 0

3 years ago