Amy and Richard each solved an equation using the quadratic formula.


Amy's Equation and Method

Latex: 4x^2+7x-20=0
4
x
2
+
7
x
−
20
=
0

Step 1: Latex: x=\frac{-7\pm \sqrt{7^2-4(1)(20)}}{2(4)}
x
=
−
7
±
√
7
2
−
4
(
1
)
(
20
)
2
(
4
)


Step 2: Latex: x=\frac{-7\pm \sqrt{49-80}}{8}
x
=
−
7
±
√
49
−
80
8


Step 3: Latex: x=\frac{-7\pm \sqrt{-31}}{8}
x
=
−
7
±
√
−
31
8


Step 4: Latex: x=\frac{-7\pm i\sqrt{31}}{8}
x
=
−
7
±
i
√
31
8


Richard's Equation and Method

Latex: x^2-6x+8=0
x
2
−
6
x
+
8
=
0

Step 1: Latex: x=\frac{6\pm \sqrt{(-6)^2-4(1)(8)}}{2(1)}
x
=
6
±
√
(
−
6
)
2
−
4
(
1
)
(
8
)
2
(
1
)


Step 2: Latex: x=\frac{6\pm \sqrt{36-32}}{2}
x
=
6
±
√
36
−
32
2


Step 3: Latex: x=\frac{6\pm \sqrt{4}}{2}
x
=
6
±
√
4
2


Step 4: Latex: x=\frac{6+\sqrt{4}}{2}
x
=
6
+
√
4
2
and Latex: x=\frac{6-\sqrt{4}}{2}
x
=
6
−
√
4
2


Step 5: Latex: x=\frac{6+2}{2}
x
=
6
+
2
2
and Latex: x=\frac{6-2i}{2}
x
=
6
−
2
i
2


Step 6: Latex: x=4
x
=
4
and Latex: x=3-i
x
=
3
−
i


Both students made a mistake.

Describe the mistake each student made.

Explain what each student needs to do to fix their mistake.

Create your own quadratic equation, and explain how to use the quadratic formula to solve it. Be specific, using Latex: a\textsf{, }b\textsf{,} and Latex: c
c
of your equation and giving the solutions to the equation you chose.

Number your responses from 1 to 3 so your instructor can tell which question you're responding to. You may not receive full credit if your teacher cannot determine that you've answered each question.

Search entries or author

The slope of the line that passes through between (0, -6) and (2, 0) is: C. 3.

The slope of a line can be defined as the measure of the ratio of the vertical distance to the horizontal distance that exists between two points on a coordinate plane.

If we are given two points on a line, (x1, y1) and (x2, y2), the slope (m) is the rise/run = change in y / change in x = (y2 - y1)/(x2 - x1).

Given the following coordinates of two points as follows, (0, -6) and (2, 0), let:

(0, -6) = (x1, y1)

(2, 0) = (x2, y2)

Plug in the values into the slope formula to find the slope:

Slope (m) = (0 - (-6)) / (2 - 0)

Slope (m) = (0 + 6) / (2 - 0) [minus multiplied by minus is plus]

Slope (m) = (6) / (2)

Slope (m) = 3

Thus, the slope of the line that passes through between (0, negative 6) and (2, 0) is calculated as: C. 3.

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The expression that represents the perimeter of the rectangle is 30h - 2

The dimensions of the rectangle are given as:

Length = 7h + 3

Width = 8h - 4

The perimeter of the rectangle is given as:

P = 2 * (Length + Width)

Substitute the known values in the above equation

P =2 * (7h + 3 + 8h - 4)

Evaluate the like terms

P = 2 * (15h - 1)

Evaluate the product

P = 30h - 2

Hence, the expression that represents the perimeter of the rectangle is 30h - 2

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a) The future value, principal plus interest, with compound interest on a principal of $8,704.56 at a rate of 4% per year compounded 2 times per year over 8 years is $11,949.50.

b) The future value, principal plus interest, with compound interest on a principal of $8,704.56 at a rate of 4% per year compounded continuously over 8 years is $11,987.29.

The future value can be determined using an online finance calculator.

Data and Calculations:

Principal (P): $8,704.56

Annual Rate (R): 4%

Compound (n): Compounding Semi-Annually

Time (t in years): 8 years

Result:

A = $11,949.50

A = P + I where

P (principal) = $8,704.56

I (interest) = $3,244.94

First, convert R as a percent to r as a decimal

r = R/100

r = 4/100

r = 0.04 rate per year,

Then solve the equation for A

A = P(1 + r/n)nt

A = 8,704.56(1 + 0.04/2)(2)(8)

A = 8,704.56(1 + 0.02)(16)

A = $11,949.50

Using the formula A = Pert

Principal (P):  $8,704.56

Annual Rate (R): 4%

Compound (n): Compounding Continuously

Time (t in years):   8 years

Result:

A = $11,987.29

A = P + I where

P (principal) = $8,704.56

I (interest) = $3,282.73

First, convert R as a percent to r as a decimal

r = R/100

r = 4/100

r = 0.04 rate per year,

Then solve the equation for A, using the mathematical constant, e = 2.71828

A = Pert

A = 8,704.56(2.71828)(0.04)(8)

A = $11,987.29

Thus, while the future value of $8,704.56 at a rate of 4% per year compounded semiannually over 8 years is $11,949.50,  the future value of $8,704.56 at a rate of 4% per year compounded continuously over 8 years is $11,987.29.

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Answer:

[tex]2x+5y=41[/tex]

Step-by-step explanation:

Median of a triangle:  A line segment that connects a vertex of a triangle to the midpoint of the opposite side.

Vertex:  The point where any two sides of a triangle meet.

Given vertices of a triangle:

Find the midpoint of BC (Point D) by using the Midpoint formula.

Midpoint between two points

[tex]\textsf{Midpoint}=\left(\dfrac{x_2+x_1}{2},\dfrac{y_2+y_1}{2}\right)\quad \textsf{where}\:(x_1,y_1)\:\textsf{and}\:(x_2,y_2)\:\textsf{are the endpoints}}\right)[/tex]

Define the endpoints:

Substitute the defined endpoints into the formula:

[tex]\textsf{Midpoint of BC}=\left(\dfrac{-21-33}{2},\dfrac{25+13}{2}\right)=(-27,19)[/tex]

Therefore, D = (-27, 19).

Find the slope of the median (line AD) using the Slope formula.

Define the points:

Substitute the defined points into the Slope formula:

[tex]\implies \textsf{slope}\:(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{19-9}{-27-(-2)}=-\dfrac{2}{5}[/tex]

Therefore, the slope of the median is -²/₅.

Substitute the found slope and one of the points into the Point-slope formula to create an equation for the median.

[tex]\implies y-y_1=m(x-x_1)[/tex]

[tex]\implies y-9=-\dfrac{2}{5}(x-(-2))[/tex]

Simplify and rearrange the equation so it is in standard form Ax+By=C:

[tex]\implies 5(y-9)=-2(x+2)[/tex]

[tex]\implies 5y-45=-2x-4[/tex]

[tex]\implies 2x+5y-45=-4[/tex]

[tex]\implies 2x+5y=41[/tex]

Therefore, the equation of the median is:  

2x + 5y = 41

Peter:

£2.30 ÷ 6 = £0.38 each sausage

Paul:

£3.50 ÷ 10 = £0.35 each sausage

As you can tell, it is cheaper to buy from Paul because each sausage is £0.35 which is 3 pence less than buying from Peter.

In other words, the best value for money would go to the 10 pack sausage, saving £0.03 per unit of sausage.

However, that doesn't answer the question. You need to find how many packages you can get with your available £10.

£10 ÷ £2.30 = 4.34...

So, you can buy 4 packs of sausage from Peter - meaning you will get 24 sausages in total. This is because each pack contains 6 sausages, and you have 4 of this. (6x4=24)

£10 ÷ £3.50 = 2.85...

So, you can buy 2 packs of sausages from Paul & meaning you will get 20 sausages in toal because each package contains 10 sausages. (2x10 = 20)

Thus, you should buy from Peter because you get 24 sausages, instead of 20 sausages from Paul, in accordance to the £10 you can spend in both shops.

Hope this helps!

we conclude that the store where you can get more sausages is in Peter the Butcher.

You have £10.

Peter the Butcher sells 6 sausages for £2.30

The number of packages that you can buy is:

£10/£2.30 = 4.3

Rounding down to the next whole number, we conclude that here you can buy 4 packages, then:

4*6 = 24 sausages

4*£2.30 = £9.20

So here you can buy 24 sausages for £9.20

In Paul the butcher, you can buy 10 sausages for £3.50

Notice that here you can only buy 2 packages (because the cost of 3 packs would be  £10.50, which is more than the amount you can spend).

Then here you only can get 20 sausages.

Then, we conclude that the store where you can get more sausages is in Peter the Butcher.

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So, the equation is sin(x + y)/sin(x - y) = (1 + cotxstany)/(1 + cotxtany)

The question has to to with trigonometric identities?

Trigonometric identities are equations that show the relationship between the trigonometric ratios.

Given the equation sin(x + y)/sin(x - y)

Using the trigonometric identities.

So, sin(x + y)/sin(x - y) = (sinxcosy + cosxsiny)/(sinxcosy + cosxsiny)

Dividing the rnumerator and denominator of ight hand side by sinx, we have

sin(x + y)/sin(x - y) = (sinxcosy + cosxsiny)/sinx/(sinxcosy + cosxsiny)/sinx

sin(x + y)/sin(x - y) = (sinxcosy/sinx + cosxsiny/sinx)/(sinxcosy/sinx + cosxsiny/sinx)

= (cosy + cotxsiny)/(cosy + cotxsiny) (since cosx/sinx = cotx)

Dividing the numerator and denominator of the right hand side by cosy, we have

= (cosy + cotxsiny)/cosy/(cosy + cotxsiny)/cosy

= (cosy/cosy + cotxsiny/cosy)/(cosy/cosy + cotxsiny/cosy)

= (1 + cotxstany)/(1 + cotxtany)   [since siny/cosy = tany]

So,  sin(x + y)/sin(x - y) = (1 + cotxstany)/(1 + cotxtany)

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By solving the equations x+2y=5, 4x-12y=-20 through elimination or substitution method we will get the value of x=1 and y=2.

Given two equations x+2y=5 and 4x-12y=-20.

We are required to solve the equations through elimination and substitution method.

Firstly we will solve the equations through elimination method.

x+2y=5-------------1

4x-12y=-20--------2

Multiply the equation 1 by 4 and then subtract the equation 2 from equation 1.

4x+8y-4x+12y=20+20

20y=40

y=2

Use the value of y in 1

x=5-2*2

x=5-4

x=1

From substitution method.

First find the value of x from first equation in terms of y.

x=5-2y-----4

Now put this value in second equation to get the value of y.

4(5-2y)-12y=-20

20-8y-12y=-20

20-20y=-20

-20y=-40

y=2

Put the value of y in equation 4.

x=5-2y

x=5-2*2

x=5-4

x=1

Hence by solving the equations x+2y=5, 4x-12y=-20 through elimination or substitution method we will get the value of x=1 and y=2.

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Answer:

2

Step-by-step explanation:

Answer:

c

Step-by-step explanation:

The measure of the angle ∠CRQ is 103° and the measure of the angle ∠CRS will be 103°.

The angle is the distance between the intersecting lines or surfaces. The angle is also expressed in degrees. The angle is 360 degrees for one complete spin.

Supplementary angle - Two angles are said to be supplementary angles if their sum is 180 degrees.

Corresponding angle - If two lines are parallel then the third line. The corresponding angles are equal angles.

In the figure, lines BD and QS are parallel. Two parallel lines B D, and Q S are intersected by a line AT at C, and R respectively such that the angle ∠DCR is 77 degrees.

We know that the angle ∠DCR & ∠SRT and ∠DCR & ∠SRT are corresponding angles. Then they are equal to each other.

∠DCR = ∠SRT = 77°

∠CRQ = ∠CRS

We know that the angle ∠CRQ and ∠CRS are supplementary angles. Then we have

∠CRQ + ∠SRT = 180°

∠CRQ + 77° = 180°

∠CRQ = 103°

Then the measure of the angle ∠CRQ is 103° and the measure of the angle ∠CRS will be 103°.

The diagram is given below.

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Answer:

Step-by-step explanation:

12 + 2x + 24 = 27 + x

2x + 36 = 27 + x

x + 36 = 27

x = -9

2(-9) + 24 = -18 + 24 = 6

12 + 6 = 18

27 + (-9) = 18

Answer:

[tex]-3(x-2)^2+5[/tex]

Step-by-step explanation:

First we can factor a -3 from the [tex]x^2[/tex] term and the [tex]x[/tex] term to get [tex]-3(x^2-4x)-7[/tex].

Then we want the stuff in the parentheses to have the form of [tex](x+b)^2[/tex], or equivalently, [tex](x^2+2b+b^2)[/tex] . So we can let [tex]2b = -4[/tex]. By solving it, we get [tex]b = -4/2 = -2[/tex]. Then our [tex]b^2[/tex] term should be [tex]b^2 = (-2)^2 = 4[/tex].

In order to make our [tex]b^2[/tex] term appear in the parentheses, we need to add and subtract our [tex]b^2[/tex] term, so we get [tex]-3(x^2 - 4x + 4 - 4) -7[/tex].

What we to keep inside our parentheses is [tex](x^2 - 4x + 4)[/tex] , so we can factor the [tex]-4[/tex] out of parentheses to get [tex]-3(x^2-4x+4-4)-7 = -3(x^2-4x+4)+(-3)(-4) - 7 = -3(x^2 - 4x + 4) + 12 - 7 = -3(x^2 - 4x + 4) + 5[/tex]

Finally, plugging  [tex]b = -2[/tex]  that we computed earlier into the equation [tex](x^2+2b+b^2) = (x+b)^2[/tex], we get  [tex](x^2 - 4x + 4) = (x-2)^2[/tex].

So we have [tex]-3(x^2-4x+4)+5 = -3(x-2)^2+5[/tex].

In summary, the procedure is

[tex]-3x^2+12x-7 \\= &-3(x^2-4x) -7 \\= &-3(x^2-4x+4-4)-7 \\= -3(x^2-4x+4) + (-3)(-4)-7\\=-3(x^2-4x+4)+12-7\\= -3(x^2-4x+4)+5 \\= -3(x-2)^2+5[/tex]

Answer:

376.99112cm²

Step-by-step explanation:

A=2 pi rh + 2 pi r²

A=(2 x pi x 5 x 7) + (2 x pi 5²)

pi 22/7

Answer:

[tex]4 < x < 20[/tex]

Step-by-step explanation:

By the triangle inequality theorem, the sum of the length of the two smaller sides must be greater than the length of the largest side. If 12 cm was the length of the largest side, notice that [tex]8+x > 12 \Rightarrow x > 4[/tex]. Similarly, if x was the length of the largest side, we'd get the inequality [tex]12+8 > x \Rightarrow 20 > x[/tex]. Combining these inequalities, we get [tex]\boxed{4 < x < 20}[/tex].

Answer:

51,840

Step-by-step explanation:

The format of P(n, r) is used for permutations,

where      [tex]P(n, r) = \frac{n!}{(n - r)!}[/tex]

To evaluate this expression, P(9,2), we use the formula of permutation,

We have the value of n = 9 and r = 2, thus,

[tex]P(n, r) = P(9, 2) = \frac{9!}{(9 - 2)!} =\frac{9!}{7!} =\frac{9*8*7!}{7!} = 9 *8 =72\\[/tex]

To evaluate this expression, P(6, 5), we use the formula of permutation,

We have the value of n = 6 and r = 5, thus,

[tex]P(n, r) = P(6, 5) = \frac{6!}{(6 - 5)!} =\frac{6!}{1!} =\frac{6*5*4*3*2*1!}{1!} = 6*5*4*3*2 = 720[/tex]

Therefore the product of P(9, 2) · P(6, 5) is

72 · 720 = 51,840

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Answer:

Slope: [tex]-\frac{1}{5}[/tex]   Y-intercept: (0,0)

Step-by-step explanation:

By putting this into a graph you can see that the y-intercept is (0,0). Also, by looking at the chart and using the walk, rise rule you can see that the slope is [tex]-\frac{1}{5}[/tex].

Answer:

4,928 rounded to the nearest hundred = 4,900

Step-by-step explanation:

Greetings!

Hope it helps!

Answer:

The answer is 4900, because the next number is less than five, due to that we don't have to add a number (+1).

---

Thank you so much for participate in the community of Brainly.com

I hope help you :)

Kind regards, Antonio.

Answer:

480

Step-by-step explanation:

1/5 of 400 =   1/5 * 400 = 80

  increase 400 by 80   = 480

Answer:

$15666.83 (2dp)

Step-by-step explanation:

Mean = Total of all values / Number of Values

= [tex]\frac{17484+14978+13521+14500+18540+14978}{6}[/tex]

=[tex]\frac{94001}{6}[/tex]

= $15666.83 (2dp)

The sequence of transformation carried out on ABCD is; B: Reflection about the y-axis followed by a translation left by 2 units

As in rotation, reflection and translation the transformed image is congruent to the original figure (i.e all the points undergo same change). Thus, analyzing any one point will give the series of transformation undertaken.

Taking point A(3, 1) into consideration; Reflecting a point (x, y) over y axis, it becomes (-x, y).

Thus, reflecting A(3, 1) over y - axis gives the the new coordinates as (-3, 1)

Now, translating a point (x, y) k units left, gives a new coordinate (x - k, y).

Thus, translating point (-3, 1) 2 units left, gives the new coordinates as A'(-5, 1).

This is the coordinates for A'.

Thus, we can conclude by saying that the sequence of transformation carried out on ABCD is reflected about the y-axis followed by a translation left by 2 units.

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[tex]g(x)=f(4x)=(4x)^2 = 16x^2[/tex]

The graph is shown in the attached image.

The linear function whose points are given is f(x)=[tex]x/5 +1[/tex]

Given the points of the function f(-10)=-1,f(0)=1,f(10)=3.

We are required to find the function whose points are given.

Function is relationship between two or more variables that are expressed in equal to form. In function each value of x must have a corresponding y value.

x value is known as domain and y value if known as codomain of the function.

Take two points (-10,-1) and (0,1) and we can form one equation as under:

y+1=(1+1/0+10) * (x+10)

y+1=2/10 (x+10)

y+1=x/5 +10/5

y+1=x/5+2

y=x/5+2-1

y=x/5+1

Hence the linear function whose points are given is [tex]f(x) = x/5 +1[/tex]

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hope this helps

by

aman10we

The value of x after solving this equation is 2

Polynomial is made up of two terms, namely Poly (meaning “many”) and Nominal (meaning “terms.”). A polynomial is an expression composed of variables, constants, and exponents, combined using mathematical operations such as addition, subtraction, multiplication, and division (No division operation by a variable). Based on the number of terms present in the expression, it is classified as monomial, binomial, and trinomial. For example P(x) = x2-5x+11

Given here, the equation as : (4x - 5)^4 = 81.​

(4x - 5)^4 = 3⁴.​

4x - 5    = 3

      x     = 2

Hence, the value of x is equal to 2

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Answer:

mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size.

Answer:

Answer:

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

We know that diagonals of rhombus are perpendicular to each other.

Hence its area is half the product of diagonals.

The diagonals are JL and KM and one of them is vertical and the other one horizontal since x- or y-coordinates are equal in pairs.

Let's find the length of diagonals, using the difference of coordinates:

Now find the area:

The value of the probability P(A and B) is 0.20

The given parameters about the probability are

P(A or B) = 0.9

P(A) = 0.5

P(B) = 0.6

To calculate the probability P(A and B), we use the following formula

P(A and B) = P(A) + P(B) - P(A or B)

Substitute the known values in the above equation

P(A and B) = 0.5 + 0.6 - 0.9

Evaluate the expression

P(A and B) = 0.2

Hence, the value of the probability P(A and B) is 0.20

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Answer:

For example, 8f + 4g is one of the most used expression

Another example is 4 (2f + g), is correct too

Answer:

scalene

Step-by-step explanation:

hidhihihihihihi

The solution to the original equation is 1, 1/6

Equations are expressions separated by mathematical operations.

Given the equation below

x/x − 1 = 6x + 1/x − 1

From the given expression, the least common denominator is x -1

Multiply both sides by x-1 to have;

x = 6x(x-1) +1

Expand

x = 6x^2-6x + 1

Equate to zero

6x^2-6x-x + 1 = 0

6x^2-7x +1= 0

The resulting quadratic equation in general form is 6x^2-7x +1 = 0

Factorize

6x^2 -6x-x + 1 = 0

Group the result

6x(x-1)-1(x-1) = 0

(6x-1)(x-1) = 0

6x - 1 = 0 and x -1 = 0

x = 1 and 1/6

Hence the solution to the original equation is 1, 1/6

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Answer:

the price of one pound of trail mix is $3

the price of one pound of jelly beans is $2.5

Step-by-step explanation :

let t be the price of one pound of trail mix.

and b be the price of one pound of jelly beans.

This statement:

“For 3 pounds of trail mix and 8 pounds of jelly beans,

the total cost is $29”

Can be translated mathematically like this :

3t + 8b = 29

This statement:

“For 5 pounds of trail mix and 2 pounds of jelly beans,

the total cost is $20”

Can be translated mathematically like this :

5t + 2b = 20

We obtain a system of two equations :

3t + 8b = 29

5t + 2b = 20

Solving the system:

3t + 8b = 29  (equation 1)

5t + 2b = 20. (equation 2)

3t + 8b = 29

20t + 8b = 80   [4×(equation 2)]

3t + 8b = 29

17t = 51  [4×(equation 2) - equation 1]

3t + 8b = 29

t = 51/17 = 3

9 + 8b = 29

t = 3

8b = 20

t = 3

b = 20/8 = 5/2

t = 3

b = 5/2

t = 3

Therefore ,the solution to our system is (3 , 2.5)

The inverse equation of x = y is y = x

Inverse equations are the opposite of an original equation

The equation is given as

x = y

Swap the positions of x and y in the above equation

y = x

This means that the inverse equation of x = y is y = x

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