Calculus

x^(2) - 10x + 16 = 0

Factors of '16', which are, 1,2,4,8,16. Select two numbers from this list that add/subtract to '10' . They are '2' and '8'.

Setting up brackets

( x 2)(x 8)

Since '16' is positive(+), then both signs must be the same. Since '10x' is negative (-) , then both signs are negative.

Hence

(x - 2)(x - 8 ) = 0

When

x - 2 = 0 x = 2

& when

x - 8 = 0

x = 8

So the two answers are ,2, & ,8, .

If x = 2 , then it is a vertical line, parallel with the y -axis.

The gradient is described as 'undefined'.

3y^(4)- y^(2) = 0

Factor

y^(2) [ 3y^(2) - 1] = 0

Hence

y^(2) = 0

It follows that y = 0

And

3y^(2) - 1 = 0

y^(2) = 1/3

y = +/- sqrt( 1/3) = +/-0.577350269.....

Hence y = 0 or y = +/-0.577350269.... to make the eq'n equal to zero.

4x - y = 1

3x + y = 13

Simultaneous Equations.

Add ; This will eliminate 'y'

Hence

7x = 14

x = 2

When x = 2

3(2) + y = 13

6 + y = 13

y = 13 - 6

y = 7

Hence the solution in ( x,y) form is ( 2, 7).

To solve a proportion with a missing variable, you can use cross-multiplication. A proportion means two ratios are equal. For example, if

3

/

4

=

𝑥

/

8

3/4=x/8, multiply across the fractions:

3

×

8

=

4

×

𝑥

3×8=4×x. This gives

24

=

4

𝑥

24=4x. Then divide both sides by 4 to find

𝑥

=

6

x=6. This simple method works for most proportion problems and helps students find the missing value quickly. For step-by-step explanations of problems like this, many students use Sorry Teacher to understand the concept more clearly.

Simultaneous Equations.

4x + 5y = 51

4x - 5y = 21

eliminate 'y' by addition.

Hence

8x = 72

x = 72/8

x = 9

When x = 9 Substitute into eith equation.

4(9) + 5y = 51

36 + 5y = 51

5y = 51 - 36

5y = 15

y = 3

Hence the answer is ( x,y) = ( 9,3).

-38n = 46

Divide both sides by ' -38'

Hence

n = 46/ -38

Cancel dowen by '2'

n = 23/ -19

n = -1.210526316.... ( Note the answer is negative)/.

2x + 2 = x + 5

Subtract '2' from both sides

Hence

2x = x + 3

Subtract 'x' from both sides

x = 3 The answer !!!!!

2x + 8 = 2(x+4)

Regional integration can take several forms, including free trade areas, customs unions, common markets, and economic unions. Free trade areas eliminate tariffs between member countries, while customs unions also adopt a common external tariff. Common markets allow for the free movement of goods, services, labor, and capital, and economic unions further integrate economies through shared policies and regulations. Each type represents varying levels of economic cooperation and political alignment among member states.

Yes, both institutions and individuals can engage in derivative market transactions. Institutions, such as banks and hedge funds, often use derivatives for hedging risk or speculation. Individuals can also participate in derivatives trading, typically through brokerage accounts that offer access to options and futures markets. However, it's essential for both parties to understand the risks involved, as derivatives can lead to significant gains or losses.

5x - 3 + x - 9 = 5x

Bring like terms together of the left side: 5x + x - 3 - 9 = 5x

Combine like terms: 6x - 12 = 5x

Add 12 to both sides: 6x = 5x + 12

Subtract 5x from both sides: x = 12

3x^(2) +9x - 2x -6

Collect 'like terms'.

Hence

3x^(2) + 7x - 6

Next write down all the factors of '3' and '6'

Hence

3 ; 1' & 3'

6 ; 1,6 ; 2,3.

From these pairs of number we select a pair from each coefficient, that add/multiply to '7' .

Hence

(3' x 3 ) & (1' x 2) ; NB 'dashes' (') to indicate source of numbers.

Write up brackets

(3x 2)(x 3) -2)(x + 3)

Next we notice that the '6' is negative, so the two signs are different (+/-) The '7x' is positive , so the larger number takes the positive sign .

Hence

(3x - 2)(x + 3)

(18x^(2) - 3x)

'3x' is a common factor to both terms , so 'take it out'.

Hence

3x(6x - 1)

Fully factored.

44/110

Cancel down by '11'

4/10

Cancel down by'2'

2/5 The simplest form

0.75 = 0.75/1.00

Cancel down the decimal point

075/100

Drop the prefix '0'

75/100

Cancel down by '25'

3/4 as a fraction in its simplest form.

-47

x times 1 is x. Anything times 1 is still that same anything.

That is 2.333 ounces

Undefined: You cannot divide by zero

Dear Questioner its not able to solve two variables for one equation, give one more equation. Thanks

5

In derivative questions, the value of zero is often used as a reference point for evaluating the rate of change of a function at a specific point. By setting the change in the variable (Δx) to zero, we can analyze the instantaneous rate of change at that point, which is the essence of the derivative. This approach allows us to find the slope of the tangent line to the curve at a given point, providing important information about the function's behavior.

2x - 3y = 2

3x + 2y = 3

Simultaneous Equations.

Arbitrarily decide which letter to eliminate.

In this case 'y'.

To do this we bring the coefficients of 'y' to the same value. . To do this multiply the top equation by '2' and the bottom eq'n by.3.

Hence

4x - 6y = 4

9x + 6y = 9

We now add the two equations. The addition of '-6y + 6y = 0 ', So 'y' is eliminated.

Hence

13x = 13

x = 1

To find 'y' , we substitute 'x = 1' into either equation.

Hence

4(1) - 6y = 4

4 - 6y = 4

-6y = 0

y = 0

So the answer in ( x,y ) form is ( 1,0 ) .

-3 + 3y = 2y + 5

Add '3' to both sides

Hence

3y = 2y + 8

Subtract '2y' from both sides.

y = 8 The answer!!!!!