through: (2,2), parallel to y=x+42)

Answer:

[tex]\frac{1}{2}[/tex] x²

Step-by-step explanation:

let the length of the side be l

Then using Pythagoras' identity on the right triangle formed by the diagonal ( hypotenuse ) and the 2 sides

The square on the hypotenuse is equal to the sum of the squares on the other 2 sides, that is

l² + l² = x²

2l² = x² ( divide both sides by 2 )

l² = [tex]\frac{1}{2}[/tex] x² ( since A of square = l² )

Answer:

Miles he drove on highway = 180 miles

Miles he drove in the city = 217 miles

Step-by-step explanation:

Lets assume that gallon used on highway = x

Miles driven on highway = y

(I) 36 miles per gallon on the highway.

36x = y    equation 1

(II) 31 miles per gallon in the city.He recently drove 397 miles on 12 gallons of gasoline

31(12-x)= 397-y    equation 2

372-31x= 397-y

Combine the constants:

-31x= 397-372-y

y-31x = 25

Now substitute the value of equation 1 in equation 2

y-31x=25

36x-31x=25

5x=25

Now divide both sides by 5

5x/5=25/5

x=5

Now substitute the value x=5 in equation 1

36x=y

36(5)=y

180= y

Now subtract the miles driven on highway from recently drove miles to get the miles driven in the city.

397-y = 389 - 180

= 217

Therefore,

Miles he drove on highway = 180 miles

Miles he drove in the city = 217 miles ....

Julio drove 0 miles on the highway and 2856 miles in the city.

To find out how many miles Julio drove on the highway and in the city, we can set up a system of equations. Let x represent the number of miles driven on the highway and y represent the number of miles driven in the city. We know that the total distance driven is 397 miles and the total amount of gas used is 12 gallons. Using the given miles per gallon ratings, we can set up the following equations:

x + y = 397

36x + 31y = 12

From the first equation, we can isolate x:

x = 397 - y

Substituting this value of x into the second equation, we can solve for y:

36(397 - y) + 31y = 12

14292 - 36y + 31y = 12

-5y = -14280

y = 2856

Now we can substitute the value of y back into the first equation to find x:

x + 2856 = 397

x = 397 - 2856

x = -2459

However, since we cannot have negative miles, we know that Julio did not drive any miles on the highway.

Therefore, Julio drove 0 miles on the highway and 2856 miles in the city.

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Answer:

x = 40 deg

Step-by-step explanation:

Given that the line at the base of the triangle is a continuous straight line,

x + (4x-20) = 180 degrees

x + 4x  - 20 = 180

5x = 180 + 20

5x = 200

x = 40 deg

Your answer is a right triangle, reason is because they usually use it for right triangles, not obtuse nor acute. Tangent ratio is used to find the length for the right triangle sides and it also gives the degree for each right triangle angle (right triangle has three angles, where there are 2 angles and 1 right angle.)

Hope this helped!

Nate

Answer:

The tangent ratio is used for right  triangles.

Step-by-step explanation:

We have been given an incomplete statement. We are supposed to fill in the given blank for statement using correct option.

Statement:

The tangent ratio is used for _  triangles.

We know that tangent is a trigonometric ratio, which represent relation between opposite side of right triangle to its adjacent side.

Therefore, the correct term for our given statement is 'right' and option C is the correct choice.

Shown below

No graph has been plotted, but the question is answerable either way and I'll be happy to help you. In this problem, we have the following inequality:

[tex]x-y-2\geq 0[/tex]

Before we focus on getting the shaded region, let's graph the equation of the line:

[tex]x-y-2=0[/tex]

So let's write this equation in slope intercept form [tex]y=mx+6[/tex]:

STEP 1: Write the original equation.

[tex]x-y-2=0[/tex]

STEP 2: Subtract -x from both sides.

[tex]x-y-2-x=0-x \\ \\ \\ Group \ like \ terms \ on \ the \ left \ side: \\ \\ (x-x) - y-2=-x \\ \\ The \ x's \ cancel \ out \ on \ the \ left: \\ \\ -y-2=-x[/tex]

STEP 3: Add 2 to both sides.

[tex]-y-2+2=-x+2 \\ \\ -y=-x+2[/tex]

STEP 4: Multiply both sides by -1.

[tex](-1)(-y)=(-1)(-x+2) \\ \\ y=x-2[/tex]

So, [tex]m=1[/tex] and [tex]b=-2[/tex]. The graph of this line passes through these points:

[tex]If \ x=0 \ then: \\ \\ y=x-2 \therefore y=(0)-2 \therefore y=-2 \\ \\ Passes \ through \ (0,-2) \\ \\ \\ If \ y=0 \ then: \\ \\ y=x-2 \therefore 0=x-2 \therefore x=2 \\ \\ Passes \ through \ (2,0)[/tex]

By plotting this line, we get the line shown in the first figure below. To know whether the shaded region is either above or below the graph, let's take point (0,0) to test this, so from the inequality:

[tex]x-y-2\geq 0 \\ \\ Let \ x=y=0 \\ \\ 0-0-2\geq 0 \\ \\ -2\geq 0 \ False![/tex]

Since this statement is false, then the conclusion is that the region doesn't include the origin, so the shaded region is below the graph as indicated in the second figure below. The inequality includes the symbol ≥ so this means points on the line are included in the region and the line is continuous.

This question is based on the slide reflection. Therefore, the correct option is (2), that is, translate 5 units down, and then reflect over the y-axis.

Given:

Reflect the triangle across the y-axis, and then translate the image 5 units down.

We have to choose the correct option as per given question.

According to the question,

It is a slide reflection.  

In this case, the order of sliding and reflection is not necessary, if the gliding is parallel to the line of reflection.

Hence, the equivalent to translate five units down, and then reflect over the y-axis.

In general,  that rotations will not be equivalent to an odd number of reflections.

Therefore, the correct option is (2), that is, translate 5 units down, and then reflect over the y-axis.

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Answerits not

Step-by-step explanation:

Answer:

Step 1: 5

Step 2: -1/5

Step 3: 2/3

Step 4: 2/3

Only one pair of opposite sides is parallel

Step-by-step explanation:

Answer:

24

Step-by-step explanation:

The area of the given figure ABCD with respective coordinates is

24 square units.

The area of a quadrilateral is nothing but the region enclosed by the sides of the quadrilateral.

Given the coordinates of the quadrilateral as; A(−4, 1), B(2, 1), C(2, −5), and D(−4, −3).

By inspection, we see that the y coordinates of A and B are the same. Thus, their length will be the difference of their x coordinates. Thus;

AB = 2 - (-4)

AB = 6

Similarly, B and C have same x coordinates. Thus;

BC = -5 - 1 = -6

A and D have same x coordinate and as such;

AD = -3 - 1 = -4

AB and BC are perpendicular to each other because of opposite signs of same Number and since AD has a different length, then we can say that the figure ABCD is a rectangle.

Thus;

Area of figure = 6 × 4 = 24 square units.

The area of the given figure ABCD with respective coordinates is

24 square units.

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Answer:

75 cubic centimeters

Step-by-step explanation:

Volume of a cone is the area of the base times a third of the height.

The base is a circle.

The formula for the area of a circle is [tex]\pi \cdot r^2[/tex].

We are given that we want to use [tex]3.14[/tex] for [tex]\pi[/tex] and

r=(diameter)/2=6/2=3 cm.

So the area of the base is [tex]3.14 \cdot 3^2=28.26[/tex].

Now the height of the cone is 8 cm.

A third of the height is 8/3 cm.

So we want to compute area of base times a third of the height.

Let's do that:

[tex]\frac{8}{3} \cdot 28.26[/tex]

75.36 cubic centimeters

To the nearest cubic centimeters this is 75

To calculate the volume of a cone, we use the formula for the volume of a cone, which is:
\[ V = \frac{1}{3} \pi r^2 h \]
Where:
- \( V \) is the volume of the cone
- \( \pi \) is the constant Pi (approximated as 3.14)
- \( r \) is the radius of the cone's base
- \( h \) is the height of the cone
Given that the diameter of the cone is 6 cm, we find the radius by dividing the diameter by 2:
\[ r = \frac{diameter}{2} = \frac{6 cm}{2} = 3 cm \]
Now we have the radius and the height (which is given as 8 cm), we can substitute these values into the formula:
\[ V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \times 3.14 \times (3 cm)^2 \times 8 cm \]
First, square the radius:
\[ (3 cm)^2 = 9 cm^2 \]
Now, perform the multiplication:
\[ V = \frac{1}{3} \times 3.14 \times 9 cm^2 \times 8 cm \]
\[ V = 3.14 \times 3 cm^2 \times 8 cm \]
\[ V = 9.42 cm^2 \times 8 cm \]
\[ V = 75.36 cm^3 \]
Finally, you want to round the volume to the nearest cubic centimeter. Since \( 75.36 \) is already a decimal to the nearest hundredth and has no fractional part in cubic centimeters, we simply round it to the nearest whole number:
\[ V \approx 75 \]
So, the volume of the cone is approximately \( 75 \) cubic centimeters.

Answer:

22.5°

Step-by-step explanation:

We are given that a bike has 16 spokes which are spaced evenly apart.

We are to find the measure of angles of rotation.

Spoke is basically the rod which comes from the center of the tire, connecting it to the round surface.

Since the total measure of angles for a circle is 360° and 16 spokes are places evenly, so each angle of rotation = [tex]\frac{360}{16}[/tex] = 22.5°

Answer:

correct answer is option D

Step-by-step explanation:

assume the speed of the unicorn for the first time to cover 50 mile be 'x'

we know,

distance = speed × time

    50  =  x  × time

     t₁ = 50 / x...............................(1)

when unicorn travel 300 mile with speed of '3x'

distance = speed × time

300  = 3 x  × time

    t₂ = 100/ x...............................(2)

dividing equation (2)/(1)

[tex]\dfrac{t_2}{t_1}  =  \dfrac{100/x}{50/x}[/tex]

    t₂ = 2 × t₁

hence, the time will be twice the first one.

correct answer is option D

Answer:

Step-by-step explanation:

The slope-intercept form of an equation of a line:

[tex]y=mx+b[/tex]

m - slope

b - y-intercept

The formula of a slope:

[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]

================================================

We have the points (-6, 4) and (2, 0).

Substitute:

[tex]m=\dfrac{0-4}{2-(-6)}=\dfrac{-4}{8}=-\dfrac{1}{2}[/tex]

Put the value of the slope and coordinates of the point (2, 0) to the equation of a line:

[tex]0=-\dfrac{1}{2}(2)+b[/tex]

[tex]0=-1+b[/tex]          add 1 to both sides

[tex]1=b\to b=1[/tex]

The equation of a line in the slope-intercept form:

[tex]y=-\dfrac{1}{2}x+1[/tex]

Convert to the standard form [tex]Ax+By=C[/tex]

[tex]y=-\dfrac{1}{2}x+1[/tex]           multiply both sides by 2

[tex]2y=-x+2[/tex]             add x to both sides

[tex]x+2y=2[/tex]

Answer:

Option 1.

Step-by-step explanation:

It is given that the line passes through the points (-6,4) and (2,0).

If a line passes through two points, then the equation of line is

[tex]y-y_1=\dfrac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]

Using the above formula the equation of line is

[tex]y-(4)=\dfrac{0-4}{2-(-6)}(x-(-6))[/tex]

[tex]y-4=\dfrac{-4}{8}(x+6)[/tex]

[tex]y-4=\dfrac{-1}{2}(x+6)[/tex]

Muliply both sides by 2.

[tex]2y-8=-x-6[/tex]

[tex]x+2y=-6+8[/tex]

[tex]x+2y=2[/tex]

Therefore, the correct option is 1.

Answer:

No. It is a constant function.

Step-by-step explanation:

The function f(x) = e^2 is not an exponential functional. Rather, it is a constant function. The reason for this is that in f(x) = e^2, there is no x involved on the right hand side of the equation. The approximate value of e is 2.718281, and the approximate value of 2.718281^2 is 7.389051. This means that f(x) = e^2 = 7.389051. It is important to note that for any value of x, the value of the function remains fixed. This is because the function does not involve the variable x in it. The graph of the function will be a line parallel to the x-axis, and the y-intercept will be 7.389051. For all the lines parallel to x-axis, the value of the function remains the same irrespective of the value of x. Also, the derivative of the function with respect to x is 0, which means that the value of the function is unaffected by the change in the value of x!!!

Answer:

The input value is x=1

Step-by-step explanation:

we know that

The intersection point both graphs is the point that  produces the same output value for the two functions

Observing the graph

The intersection point is (1,1)

therefore

For x=1 (input value)

The output value for the two functions is equal to 1

Answer:

ITS X=1

Step-by-step explanation:

Answer:

100 degrees (first choice)

Step-by-step explanation:

So we can actually find the degree measure of arc KI because there is a theorem that says this twice the angle IJK.

75(2)=150 is the degree measure of arc KI.

So a full rotation around a circle is 360 degrees.

This means the following equation should hold:

m(arc)KJ+m(arc)IJ+m(arc)IK=360

Inserting the values given for the arc measures we know:

110         +m(arc)IJ+    150    =360

Add 110 and 150:

260     +m(arc)IJ                   =360

Subtract 260 on both sides:

             m(arc)IJ                   =100

So m(arc)IJ is 100 degrees.

Answer:

-3/4 degree Fahrenheit

Step-by-step explanation:

The temperature is decreasing by 3/16 each day

We write that as -3/16

There are 4 days in which we are measuring

Total temperature change = rate of change* number of days

                                            =-3/16 * 4

                                            = -12/16

We can simplify this by dividing the top and bottom by 4

-12/4 =3

16/4 =4

                                         =12/16 = -3/4

Answer:

-3/4 degree Fahrenheit

Step-by-step explanation:

If the tempature is changing every day by -3/16 degree fahrenhiet, the change in the tempature after 4 days is -3/4 degree Fahrenheit.

Answer:

k = 6

Step-by-step explanation:

i got it right on the test and i also graphed it

Please mark brainliest!

by the way when you need to graph something use a website called Desmos.  or just go onto the search bar on the top of your screen adn type in desmos. com

The transformation of a function may involve any change. The correct option is D.

The transformation of a function may involve any change.

If the original function is y = f(x), assuming the horizontal axis is the input axis and the vertical is for outputs, then:

Given that f(x) = 0.6x is replaced by the graph of g(x) = 0.6x + k. And g(x) is obtained by shifting f(x) up by 6 units. Therefore, the value of k is 6.

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Answer:

Option B - [tex]20\leq x\leq 24.60[/tex]  

Step-by-step explanation:

Given : Tyrese works each day and earns more money per hour the longer he works. Write a function to represent a starting pay of $20 with an increase each hour by 3%.

To find : Determine the range of the amount Tyrese makes each hour if he can only work a total of 8 hours.

Solution :

A starting pay is $20.

let x be the number of hours.

There is pay of $20 with an increase each hour by 3%.

i.e. Increment is [tex]\frac{3}{100}\times 20\times x=\frac{3}{5}x[/tex]

Total earnings a function can represent is

[tex]y=20+\frac{3}{5}x[/tex]

We have given, he can only work a total of 8 hours.

So, The maximum amount she make in 8 hours is

[tex]y=20+\frac{3}{5}\times 8=20+4.8[/tex]

[tex]y=24.8[/tex]

Initial amount is $20.

Therefore, The range of the amount Tyrese makes each hour if he can only work a total of 8 hours is [tex]20\leq x\leq 24.80[/tex]

So, Approximately the required result is option B.

Answer:

1 and 4

5 and 8  are alternate exterior angles

2 and 3

6 and 7 are alternate interior angles

Step-by-step explanation:

The alternate exterior angles are the angles on the outside that are opposite each other

1 and 4 are alternate exterior angles

5 and 8 are alternate exterior angles

The alternate interior angles are the angles on the inside that are opposite each other

2 and 3 are alternate interior angles

6 and 7 are alternate interior angles

Answer:

1 and 4 , 5 and 8

Step-by-step explanation:

Answer:

see explanation

Step-by-step explanation:

The equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

To calculate m use the slope formula

m = (y₂ - y₁ ) / (x₂ - x₁ )

3

Using (x₁, y₁ ) = (0, 0) and (x₂, y₂ ) = (- 3, 6)

m = [tex]\frac{6-0}{-3-0}[/tex] = [tex]\frac{6}{-3}[/tex] = - 2

Since the line passes through the origin (0, 0) then y- intercept is 0

y = - 2x ← equation of line

4

let (x₁, y₁ ) = (6, 0) and (x₂, y₂ ) = (0, 3)

m = [tex]\frac{3-0}{0-6}[/tex] = [tex]\frac{3}{-6}[/tex] = - [tex]\frac{1}{2}[/tex]

note the line crosses the y- axis at (0, 3) ⇒ c = 3

y = - [tex]\frac{1}{2}[/tex] x + 3 ← equation of line

5

let (x₁, y₁ ) = (0, 3) and (x₂, y₂ ) = (-2, - 3)

m = [tex]\frac{-3-3}{-2-0}[/tex] = [tex]\frac{-6}{-2}[/tex] = 3

note the line crosses the y- axis at (0, 3) ⇒ c = 3

y = 3x + 3 ← equation of line

Answer:

-1680

Step-by-step explanation:

Above would be positive.

Below would be negative.

You have 1680 below, so the answer as a signed number for the elevation would be -1680.

The signed number - 1680 feet addresses the excavator's exhuming point 1680 feet underneath ocean level (negative worth demonstrates beneath ocean level).

The signed number addressing the height of the point 1680 feet beneath ocean level is - 1680 feet. The negative sign demonstrates that the worth is beneath the reference point, which for this situation is ocean level.

With regard to heights, we utilize positive numbers to address positions over the reference point (ocean level) and negative numbers to address positions beneath it.

The miner dug downward in this scenario, lowering the elevation above sea level. Since it is below the reference point, the elevation is negative.

The greatness of - 1680 feet shows the separation from ocean level to the place of removal, and the negative sign demonstrates the bearing beneath ocean level.

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Answer:

ASA postulate

Step-by-step explanation:

The sum of the measures of interior angles of triangle is always 180°, so

∠LRT=180°-∠RTL-∠RLT=180°-∠2-∠4;

∠KST=180°-∠STK-∠SKT=180°-∠1-∠3.

Since

∠1≅∠2 and ∠3≅∠4, we have that ∠LRT ≅ ∠KST.

The ASA (Angle-Side-Angle) postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. (The included side is the side between the vertices of the two angles.)

In your case:

Triangles TKS and TLR are congruent by ASA postulate.

Answer:

2

Step-by-step explanation:

[tex](\frac{3}{4})^6 \times (\frac{16}{9})^5=(\frac{4}{3})^{x+2}[/tex]

[tex]\frac{3^6}{4^6} \cdot \frac{16^5}{9^5}=\frac{4^{x+2}}{3^{x+2}}[/tex]

[tex]\frac{3^6}{4^6} \cdot \frac{(4^2)^5}{(3^2)^5}=\frac{4^{x+2}}{3^{x+2}}[/tex]

[tex]\frac{3^6}{4^6} \cdot \frac{4^{10}}{3^{10}}=\frac{4^{x+2}}{3^{x+2}}[/tex]

[tex]\frac{3^6}{3^{10}} \cdot \frac{4^{10}}{4^6}=\frac{4^{x+2}}{3^{x+2}}[/tex]

[tex]3^{-4} \cdot 4^{4}=4^{x+2}3^{-(x+2)}[/tex]

This implies

x+2=4

and

-(x+2)=-4.

x+2=4 implies x=2 since subtract 2 on both sides gives us x=2.

Solving -(x+2)=-4 should give us the same value.

Multiply both sides by -1:

x+2=4

It is the same equation as the other.

You will get x=2 either way.

Let's check:

[tex](\frac{3}{4})^6 \times (\frac{16}{9})^5=(\frac{4}{3})^{2+2}[/tex]

[tex](\frac{3}{4})^6 \times (\frac{16}{9})^5=(\frac{4}{3})^{4}[/tex]

Put both sides into your calculator and see if you get the same thing on both sides:

Left hand side gives 256/81.

Right hand side gives 256/81.

Both side are indeed the same for x=2.

Answer:

if you mean volume its 840

Step-by-step explanation:

i just multiplied the 3 numbers

Answer:

[tex]-\frac{3080}{81}[/tex]

Step-by-step explanation:

The given function is:

[tex]p(x)=3x^{5}+2x^{2}-5[/tex]

We have to find the value of the function at x = -5/3

In order to do this we need to replace every occurrence of x in the given function by -5/3. i.e.

[tex]p(-\frac{5}{3})=3(-\frac{5}{3})^{5}+2(-\frac{5}{3} )^{2}-5\\\\ p(-\frac{5}{3})=3(-\frac{3125}{243} )+2(\frac{25}{9} )-5\\\\p(-\frac{5}{3})=-\frac{3125}{81}+\frac{50}{9}-5\\\\ p(-\frac{5}{3})=-\frac{3080}{81}[/tex]

Thus, the value of the function at x =-5/3 is [tex]-\frac{3080}{81}[/tex]

Answer:

4x - 3y = 5

Step-by-step explanation:

The equation of a line in standard form is

Ax + By = C ( A is a positive integer and B, C are integers )

First obtain the equation in point- slope form

y - b = m(x - a)

where m is the slope and (a, b) a point on the line

Calculate m using the slope formula

m = (y₂ - y₁ ) / (x₂ - x₁ )

with (x₁, y₁ ) = (- 1, - 3) and (x₂, y₂ ) = (2, 1)

m = [tex]\frac{1+3}{2+1}[/tex] = [tex]\frac{4}{3}[/tex]

Using (a, b) = (2, 1), then

y - 1 = [tex]\frac{4}{3}[/tex] (x - 2) ← in point- slope form

Multiply both sides by 3

3y - 3 = 4(x - 2) ← distribute and rearrange

3y - 3 = 4x - 8 ( add 8 to both sides )

3y + 5 = 4x ( subtract 3y from both sides )

5 = 4x - 3y, so

4x - 3y = 5 ← in standard form

Sure, let's find the equation of the line that passes through the points (-1, -3) and (2, 1) step-by-step.
1. First, calculate the slope (m) of the line using the formula:
  \[
  m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-3)}{2 - (-1)}
  \]
  \[
  m = \frac{1 + 3}{2 + 1} = \frac{4}{3}
  \]
2. The slope-intercept form of a line equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
  We already found the slope \( m = \frac{4}{3} \), so we just need to find \( b \).
  Using the first point (-1, -3), plug the values into the slope-intercept form:
  \[
  -3 = \frac{4}{3}(-1) + b
  \]
  Calculate \( b \):
  \[
  -3 = -\frac{4}{3} + b
  \]
  Add \( \frac{4}{3} \) to both sides:
  \[
  b = -3 + \frac{4}{3}
  \]
  \[
  b = -\frac{9}{3} + \frac{4}{3}
  \]
  \[
  b = -\frac{5}{3}
  \]
3. Now we have \( y = \frac{4}{3}x - \frac{5}{3} \) in slope-intercept form.
4. Next, we will convert this to standard form, which is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) is positive.
  Multiply both sides of the slope-intercept equation by 3, the common denominator, to eliminate fractions:
  \[
  3y = 4x - 5
  \]
5. Rewriting in standard form, we move the \( x \)-term to the left side:
  \[
  -4x + 3y = -5
  \]
6. In standard form, \( A \) should be positive. If we multiply the entire equation by -1, we will make \( A \) positive:
  \[
  4x - 3y = 5
  \]
7. This is already simplified since the greatest common divisor (GCD) of 4, -3, and 5 is 1. Thus, the coefficients are already in their simplest integer values.
The final equation for the line in standard form is:
\[ 4x - 3y = 5 \]

Answer:

D

Step-by-step explanation:

This is exponential growth, and to my understanding, the format goes:

initial amount (percent growth/ decay)^time

percent growth = (decimal percent + 1)

percent decay = (1 - decimal percent)

Your equation:

1500(1.02)^t

Using the above format, 1500 appears to be the initial amount, which increases by 2% per annum.

i think

Answer:

m=4

Step-by-step explanation:

11m - 15 -5m= 9

Combine like terms

6m -15 =9

Add 15 to each side

6m - 15+15 = 9+15

6m = 24

Divide each side by 6

6m/6 =24/6

m = 4

Answer:

[tex]\frac{x-1}{x^{2} +2x-3}[/tex]

[tex]\frac{1}{x+3}[/tex]

Step-by-step explanation:

first, subtract the two fractions

You will have

[tex]\frac{x-1}{x^{2} +2x-3}[/tex] the un-simplified difference of those expressions

Lets try to simplify that

Lets find the roots of the polynomial from the bottom using the formula for quadratic equation, (-b +/- sqrt( b2-4*a*c)) /(2*a)

Which are x=-3, and x=1

Thus, you have (x+3)*(x-1)

So

[tex]\frac{x-1}{x^{2} +2x-3} = \frac{x-1}{(x+3) (x-1)}= \frac{1}{x+3}[/tex]

Answer:

y = (x - [tex]\frac{1}{2}[/tex] )² - [tex]\frac{25}{4}[/tex]

Step-by-step explanation:

Given

y = (x + 2)(x - 3) ← expand factors

  = x² - x - 6

Use the method of completing the square

add/ subtract ( half the coefficient of the x- term )² to x² - x

y = x² + 2(- [tex]\frac{1}{2}[/tex]) x + [tex]\frac{1}{4}[/tex] - [tex]\frac{1}{4}[/tex] - 6

  = (x - [tex]\frac{1}{2}[/tex] )² - [tex]\frac{25}{4}[/tex]