Use the quadratic formula to find the solutions to the equation.
7- 3x+1=0

Answer:

P(x) =  (x-1)² (x-3)

Step-by-step explanation:

P(x)=x³- 5x² + 7x-3 = x² (x-1) - 4x(x-1) + 3(x-1) = (x²-4x+3)(x-1)

P(x) = (x-1)(x-3)(x-1) = (x-1)² (x-3)

Answer:

Part 1) [tex]BC=68\ units[/tex]

Part 2) [tex]DE=36\ units[/tex]

Part 3) [tex]CE=34\ units[/tex]

Step-by-step explanation:

we know that

The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side

so

Part 1) Find the length of  BC

Applying the Midpoint Theorem

[tex]DF=\frac{1}{2}BC[/tex]

we have

[tex]DF=34\ units[/tex]

substitute the given value

[tex]34=\frac{1}{2}BC[/tex]

solve for BC

[tex]BC=34(2)=68\ units[/tex]

Part 2) Find the length of  DE

Applying the Midpoint Theorem

[tex]DE=\frac{1}{2}AC[/tex]

we have

[tex]AC=72\ units[/tex]

substitute the given value

[tex]DE=\frac{1}{2}(72)[/tex]

[tex]DE=36\ units[/tex]

Part 3) Find the length of  CE

we know that

[tex]BC=CE+BE[/tex] ----> by addition segment postulate

The point E is the midpoint segment BC

That means

[tex]CE=BE[/tex]

so

[tex]BC=2CE[/tex]

we have

[tex]BC=68\ units[/tex]

substitute

[tex]68=2CE[/tex]

solve for CE

[tex]CE=68/2[/tex]

[tex]CE=34\ units[/tex]

Answer:

Step-by-step explanation:

Answer:

a. Fraction of games won by team is [tex]\frac{3}{4}[/tex].

b. Total games played by Katy as catcher is 17.

Step-by-step explanation:

Here is the complete question: A baseball team played 32 games and lost 8 Katy was the catcher in 5/8 of the winning games and 1/4 of the losing games.

a. What fraction of the games did the team win?

b. In how many games did Katy play catcher?

Given: Total number of games= 32

           Total games lost= 8.

           Katy was the catcher in 5/8 of the winning games.

           Katy was the catcher in 1/4 of the losing games.

Now, finding total number games that team won.

Winning total= [tex]32-8= 24[/tex]

∴ Total number of games that team won is 24.

Next finding fraction of the games did the team win.

Fraction of games won by team is [tex]\frac{Games\ won}{Total\ number\ of\ games}[/tex]

∴ Fraction of games won by team = [tex]\frac{24}{32} = \frac{3}{4}[/tex]

Fraction of games won by team is [tex]\frac{3}{4}[/tex]

Lets find out numbers of games Katy play chatcher.

We know, Katy was the catcher in 5/8 of the winning games.

∴ Katy was catcher= [tex]\frac{5}{8} \times 24= 5\times 3[/tex]

Katy was catcher in 15 of the wining games.

We also know, Katy was the catcher in 1/4 of the losing games.

∴ Katy was catcher= [tex]\frac{1}{4} \times 8= 1\times 2[/tex]

Katy was the catcher in 2 of the losing games.

Total games played by Katy as catcher= Catcher in wining games+ catcher in losing games.

∴ Total games played by Katy as catcher= [tex]15+2= 17 games.[/tex]

Hence, total games played by Katy as catcher is 17.

Answer:

Step-by-step explanation:

a spinner with the colors orange, yellow, purple, and green.....so this spinner only has 4 spaces....and 1 of those spaces is purple...so the ratio of stopping on purple is 1/4 or 0.25

The spinner has 4 colors, and purple is one of these. To find the theoretical probability of it stopping at purple, we divide 1 by 4, resulting in 0.25. Therefore, the theoretical probability of the spinner stopping on purple rounded to nearest hundredth is 0.25.

The probability of the spinner stopping on the color purple can be calculated by dividing the number of ways this can happen (which is 1, as there's only one purple on the spinner) by the total number of outcomes in the sample space (which is 4, as there are 4 colors on the spinner).

So, the probability would be 1 ÷ 4 = 0.25.

To get the value as a rounded decimal to the nearest hundredth, it stays 0.25 as it is already to the nearest hundredth.

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(-8,1) should be the correct answer! Good Luck and Happy Holidays!!

Answer: (1)  x²y (2) 7pqr  (3) a  (4) 6a²  (5) y³z²

Step-by-step explanation:

(1)  x^4y³ and x²y

     x^4y³ =  x²y(x²y²)

    x²y      = x²y

Therefore the Highest Common factor

               x²y which is the common factor.

(2) 14p²qr^4 = 2 x 7 x p² x q  x r^4

   49p²q²r    = 7 x 7 x p² x q² x r

   35pqr       = 5 x 7 x p x q x r

Therefore HCF =  7 x p x q x r

                         = 7pqr

(3) -a^5             = -( a x a x a x a x a )

                         = -(a^5)

       ab³            =   a x b x b x b

Therefore HCF =   a

(4)   6a²             = 2 x 3 x a x a

      -18a6          = -( 2 x3 x 3 x a x a x a x a x a x a)

     -12a²            = -( 2 x 2 x 3 x a x a )

Therefore HCF  = 2 x 3 x a²

                          = 6a²

(5)   3y³z²           = 3 x y x y x y x z x z , 3 x y³ x z²

      15y^4z²       =  3 x 5 x y x y x y x y x y x z x z, 3 x 5 x y^4 x z²

      y^6z³          = y^6 x z³

Therfore HCF  = y³ x z²

                         = y³z²

Answer:

The radius of the given equation is 2 .

Step-by-step explanation:

The equation of a circle whose center (h, k) and radius is r,

(x - h)² + (y - k)² = r² ..............................(i)

Here given that-

x² + y² + 21 = 10 x

x² - 10 x + y² + 21 = 0

( To making perfect square, add and subtract 25 ) -

x² - 10 x + 25 - 25 + y² + 21 = 0

(x² - 10 x + 25) + y² -4 =0

[As we know that -

(x - 5)² = x² - 10 x + 25]

(x - 5 )² + y² = 4

(x - 5)² + (y - 0)² = 4

(x - 5 )² + (y- 0)² = 2² ................................(ii)

On comparing equation (i) and(ii)-

r = 2

Hence the radius of the given equation is 2 and center is ( 5, 0).

Answer:

[tex]\frac{3}{4}[/tex]

Step-by-step explanation:

Given equation: [tex]\frac{3}{4}y + \frac{1}{2} = 3 \frac{1}{4}[/tex]

In the given equation, [tex]\frac{3}{4}[/tex] is a coefficient and y is the variable.

coefficient is a constant number or quantity multiplied to a variable in an algebric expression. Like in the above equation [tex]\frac{3}{4}[/tex] is a coefficient and y is the variable. We use term variable for y as its value may vary or change. If variable does not have any coefficient in any expression then in that case, we consider 1 as coefficient, for example: [tex]x+4[/tex], here variable have 1 as coefficient.

Hence, the fractional coefficient is [tex]\frac{3}{4}[/tex].

Final answer:

The fractional coefficient in the equation 3/4y+1/2=3 1/4 is 3/4.

In the given equation, the fractional coefficient is 3/4, which is the fraction that multiplies the variable y.

Explanation:

The fractional coefficient in the equation 3/4y+1/2=3 1/4 is 3/4.

In the equation 3/4y + 1/2 = 3 1/4, the fractional coefficient refers to the fraction that is multiplied by the variable y.

In this case, the coefficient is 3/4.

When working with algebraic expressions and equations, understanding how to identify and manipulate coefficients is essential for problem-solving.

The student's query seems to be aimed at identifying the coefficient of the variable in the equation. In algebra, fractions like 3/4 can be seen as coefficients when they are directly next to a variable.

No unit cancellations are needed here as the question pertains to pure algebra and not to unit conversions or applied mathematics.

Answer:

Step-by-step explanation:

7.56 × 10^3

= 7.56×1000

= 7560

Answer:

447,697.125

Step-by-step explanation:

multiply 7.65 by 10

76.5

76.5 to 3rd power is just 76.5 times 76.5 times 76.5

answer is 447,697.125

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Answer:

The Sum Of The Integers From -6 To 58 is 1690.

Step-by-step explanation:

Given,

[tex]a=-6[/tex]

[tex]T_n=58[/tex]

We have to find out the sum of integers from -6 To 58.

Firstly we will find out the total number of terms that is 'n'.

Here [tex]a_1=-6\ and\ a_2=-5[/tex]

[tex]\therefore d=a_2-a_1=-5-(-6)=-5+6=1[/tex]

Now we use the formula of A.P.

[tex]T_n=a+(n-1)d[/tex]

On substituting the values, we get;

[tex]58=-6+(n-1)1\\\\n-1=58+6\\\\n-1=64\\\\n=64+1=65[/tex]

So there are 65 terms in between  -6 To 58.

That means we have to find the sum of 65 terms in between  -6 To 58.

Now we use the formula of Sum of n_terms.

[tex]S_n=\frac{n}{2}(2a-(n-1)d)[/tex]

On substituting the values, we get;

[tex]S_{65}=\frac{65}{2}(2\times-6+(65-1)1)\\\\S_{65}=\frac{65}{2}(-12+64)\\\\S_{65}=\frac{65}{2}\times52\\\\S_{65}=65\times26=1690[/tex]

Hence The Sum Of The Integers From -6 To 58 is 1690.

Final answer:

The sum of the integers from -6 to 58 is calculated using the formula for the sum of an arithmetic series and is found to be 1,690.

Explanation:

To find the sum of the integers from -6 to 58, we can use the formula for the sum of an arithmetic series, which is S = n/2 * (a1 + an), where S is the sum of the series, n is the number of terms, a1 is the first term, and an is the last term.

We first need to determine the number of terms in our series. Since our series starts at -6 and ends at 58, we can find the number of terms by subtracting -6 from 58 and then adding 1 (because both ends are inclusive). That gives us 58 - (-6) + 1 = 65 terms.

Now, we can plug the values into the sum formula:

S = 65/2 * (-6 + 58) = 32.5 * 52 = 1690.

Therefore, the sum of the integers from -6 to 58 is 1,690.

Option E

The numerator of fraction is 21

[tex]2\frac{5}{8}[/tex] is to be written as improper fraction in lowest terms

To find: numerator of fraction

An improper fraction is a fraction in which the numerator (top number) is greater than or equal to the denominator (bottom number).

To convert a mixed fraction to an improper fraction, follow these steps:

[tex]2\frac{5}{8} = \frac{8 \times 2 + 5}{8} = \frac{16 + 5}{8} = \frac{21}{8}[/tex]

Thus the numerator of fraction is 21. Option E is correct

Answer:

[tex]x^{4} -18x^{3}+104x^{2} -172x-100[/tex]

Step-by-step explanation:

The 3 roots are given out of which 2 are real and 1 is imaginary. For a polynomial of least degree having real coefficients, it must have a complex conjugate root as the 4th root. Therefore, based on 4 roots, the least degree of polynomial will be 4. Finding the polynomial having leading coefficient=1 and solving it based on multiplication of 2 quadratic polynomials, we get:  

[tex]\\\\x_{1} = 2-\sqrt{6} \\x_{2} = 2+\sqrt{6} \\x_{3}=7-i \\x_{4}=7+i \\\\P(x)=1(x-x_{1})(x-x_{2} )(x-x_{3} )(x-x_{4} ) \\\\=(x-(2-\sqrt{6}))(  x-(2+\sqrt{6} )) (x-(7-i))( x-(7+i))\\=((x-2)+\sqrt{6})( ( x-2)-\sqrt{6} ) ((x-7)+i)( (x-7)-i)\\=((x-2)^{2} -(\sqrt{6} )^{2} )((x-7)^{2}-(i)^{2})\\=(x^{2} -4x-2)(x^{2} -14x+50)\\=x^{4} -18x^{3}+104x^{2} -172x-100\\[/tex]

Answer:

X^4-16x^3+73x^2-30x-250

Step-by-step explanation:

Answer:

D. Patel found the incorrect quotient and checked his work using multiplication incorrectly.

Step-by-step explanation:

The dividend is [tex]8,[/tex] the divisor is [tex]\dfrac{1}{6}.[/tex] Divide the dividend by the divisor (division by fraction means multiplication by its reciprocal):

[tex]8\div \dfrac{1}{6}=8\times \dfrac{6}{1}=8\times 6=48,[/tex]

so the quotient is 48, not [tex]\dfrac{1}{48}.[/tex]

To check the answer, Patel has to multiply the quotient by the divisor to get the dividend. Check answer:

[tex]48\times \dfrac{1}{6}=\dfrac{48}{6}=8[/tex]

Hence, Patel found the incorrect quotient and checked his work using multiplication incorrectly.

The most accurate description of Patel's work is option D. Patel found an incorrect quotient and checked his work using multiplication incorrectly. therefore, option D. Patel found an incorrect quotient and checked his work using multiplication incorrectly.

Patel initially calculated 8 divided by 1/6 as 1/48, which is incorrect.

The correct quotient is 48.

When Patel multiplied 1/48 by 6, he got 1/8, which is also incorrect.

The correct result should be 48, not 1/8.

So, Patel's division was incorrect, and his check using multiplication was also incorrect.

In summary, Patel made an error in the initial division and didn't accurately check his work using multiplication, leading to an incorrect final result in both cases.

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Answer:

She needs $8 more to buy a game that cost $45

Step-by-step explanation:

10 * 2 = 20

5 * 3 = 15

1 * 2 = 2

20+15+2 = 37

45-37 = $8

She needs $8 more to buy a game that cost $45

Answer:

A. 17 , 11

f(4)=3(4) + 5

=17

f(x)= 38

38 = 3x + 5

38 - 5 = 3x

33 = 3x

x =11

Answer:

[tex]f(1)=9[/tex]

Step-by-step explanation:

[tex]Given \ f(n+1)=f(n)\\f(3)=f(2)\\f(2)=f(1)\\f(1)=f(2)=f(3)\\f(1)=f(3)\\f(1)=9[/tex]

Answer:

36 trees were infected with the virus.

Step-by-step explanation:

Given:

Total number of trees =  90

Percentage of tree that had virus  40%

To find:

Number of tress that were infected by virus = ?

Solution:

Let the number trees that was affected by the virus be X.

then

X = 40% of 90

X = [tex]\frac{40}{100} \times 90[/tex]

X = [tex] 0.4 \times 90[/tex]

X= 36

There are probably some choices we're not being shown, but let's just do the whole procedure.

x² + 6x - 8 = 0

Step one: move the 8 to the other side.

x² + 6x = 8

Step two: add (6/2)²=9 to both sides.

x² + 6x + 3² = 8 + 9

Step three: Factor

(x + 3)² = 17

Step four: take the square root

x + 3 = ±√17

Step five: Solve for x,

x = -3  ± √17

Answer:

The cars are travelling ar the speed 45 mph.

Step-by-step explanation:

Let the two cars are moving at the speed of x mph.

Now, Car A reaches its destination in 15 minutes i.e. [tex]\frac{15}{60} = 0.25[/tex] hours.

Therefore, Car A travels in 15 minutes by 0.25x miles.

Again, Car B reaches its destination in 35 minutes i.e. [tex]\frac{35}{60} = 0.583[/tex] hours.

Therefore, Car B travels in 35 minutes by 0.583x miles.

Given that, (0.583x - 0.25x) = 15

⇒ 0.333x = 15

⇒ x = 45 mph

Therefore, the cars are travelling ar the speed 45 mph. (Answer)

The cars travel at a speed of 3/4 miles per minute.

To find the speed of the cars, we can use the formula Speed = Distance/Time. Let's assume that Car A's speed is x miles per minute. Since Car B travels 15 miles farther than Car A and takes 35 minutes to reach its destination, Car B's speed would also be x miles per minute. Now, using the given information, we can set up two equations:

x * 15 = x * 35 - 15

Simplifying the equation, we get:

15x = 35x - 15

20x = 15

x = 0.75

Therefore, the cars travel at a speed of 0.75 miles per minute or the fraction 3/4 miles per minute.

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Answer:

21 mini-muffins

Step-by-step explanation:

7 multiplied by 3 equals 21

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Tony bought a total of 21 mini-muffins. This was calculated by multiplying the number of packages he bought (7) by the number of mini-muffins in each package (3).

This question is about a mathematical operation called multiplication. Tony purchases 7 packages of mini-muffins. There are 3 mini-muffins in each package. To find the total number of mini-muffins Tony buys, multiply the number of packages he bought (7) by the number of mini-muffins in each package (3). This calculation is written as 7 * 3. Therefore, Tony buys a total of 21 mini-muffins.

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Answer:

3.5 hours

Step-by-step explanation:

Given: Distance= 26.2 miles

           Speed= 12 km/h

First lets convert the unit of speed from kmh to mph.

Remember, 1 mile= 1.609 km

∴ For speed of 12 km/h= [tex]\frac{12}{1.609}= 7.45\ mph[/tex]

Speed= 7.45 mph

Next, find time taken by Allan to complete a Marathon.

[tex]Time= \frac{Distance}{speed}[/tex]

⇒ Time= [tex]\frac{26.2}{7.45}[/tex]

∴ Time= 3.516 hours ≅ 3.5 h (nearest tenth)

∴ Allan took 3.5 hours to complete marathon.

Allan takes approximately 3.5 hours to complete a marathon. This is determined by converting the marathon distance to kilometers and dividing it by his running speed.

To determine how long it takes Allan to complete a marathon, we need to convert the marathon distance from miles to kilometers and then find the time it takes based on his running speed.

1 mile is approximately 1.60934 kilometers, so:

26.2 miles × 1.60934 km/mile ≈ 42.164 km

Allan runs at an average speed of 12 kilometers per hour. The time (t) needed to complete the marathon can be calculated as:

t = distance / speed

t ≈ 42.164 km / 12 km/h ≈ 3.514 hours

Rounding to the nearest tenth of an hour, Allan takes approximately 3.5 hours to complete the marathon.

Answer:

I'm not Certain but I'm thinking that it would be 4 boxes of the regular tea and 5 of the herbal tea.

Step-by-step explanation:

herbal- $1.50

regular- $1

$1*4 =$4

$1.50*5 = $7.50

$4.00+ $7.50=11.50

Answer:

Herbal tea box = 3

Regular tea box =  7

Step-by-step explanation:

Le x be the number of herbal tea box and y be the number of regular tea box.

The herbal tea variety cost $1.50 per box and the regular variety cost $1.00 per box they bought A total of 10 boxes and spent $11.50.

[tex]x+y=10\Rightarrow y=10-x[/tex]          .... (1)

[tex]1.50x+1.00y=11.50[/tex]             ... (2)

From (1) and (2) we get

[tex]1.50x+1.00(10-x)=11.50[/tex]

[tex]1.50x+10-1.00x=11.50[/tex]

[tex]0.50x=11.50-10[/tex]

[tex]0.50x=1.50[/tex]

[tex]x=\dfrac{1.50}{0.50}=3[/tex]

Substitute x=3 in equation (1).

[tex]y=10-x=10-3=7[/tex]

Therefore, the number of herbal tea boxes is 3 and the number of regular tea boxes is 7.

Answer:

Possible numbers for Joshua's house are #13 and #21.

Step-by-step explanation:

Given:

Vacation houses are equally spaced along a street.

The first house is marked with #10 and so on.

Maria lives in house marked as #17.

Joshua lives 4 houses away from Maria.

So, the difference between Joshua's house number and Maria's house number is 4.

Let 'x' be the house number of Joshua's house.

Case 1: If Joshua's house is after Maria's house. So, [tex]x> 17[/tex]

[tex]x-17=4\\x=17+4=21[/tex]

Case 2: If Joshua's house is before Maria's house. So, [tex]x<17[/tex]

[tex]17-x=4\\17-4=x\\13=x\ or\\x=13[/tex]

Therefore, the possible numbers for Joshua's house are #13 and #21.

Answer:

The standard form is [tex]7x^7 +6x^5-3x^3 + 8x[/tex]

Step-by-step explanation:

Given:

[tex]6x^5 + 8x-3x^3+7x^7[/tex]

To Find :

standard form of [tex]6x^5 + 8x-3x^3+7x^7[/tex]

Solution:

A polynomial is in standard form when its term of highest degree is first, its term of 2nd highest is 2nd etc.

In order to write any polynomial in standard form, you look at the degree of each term. You then write each term in order of degree, from highest to lowest, left to write.

Now lets check the degree of each term in the polynomial

The degree of 6x is 5

The degree of 8x is 1

The degree of 3x is 3

 The degree of 7x is 7

Now rewrite  the polynomial in the order of the degree, from highest to lowest

[tex]7x^7 +6x^5-3x^3 + 8x[/tex]

The total commission that will be paid during the two years is $ 225

Solution:

Given that insurance company pays its agents 40% commission on the first year’s premium and 5% on the second year’s premium for life insurance policies

The premiums are 500$ per year

Commission on first year:

40% commission on the first year’s premium

Thus 40 % commission on $ 500

[tex]\rightarrow 40 \% \text{ of } 500\\\\\rightarrow 40 \% \times 500\\\\\rightarrow \frac{40}{100} \times 500\\\\\rightarrow 200[/tex]

Thus commission paid for first year is $ 200

Commission on second year:

5 % commission on the second year’s premium

Thus 5 % commission on 500

[tex]\rightarrow 5 \% \text{ of } 500\\\\\rightarrow 5 \% \times 500\\\\\rightarrow \frac{5}{100} \times 500 = 25[/tex]

Thus commission paid for second year is $ 25

Total commission paid = $ 200 + $ 25 = 225

Thus Total commission paid is $ 225

Answer: 128.86cm²

Step-by-step explanation:

The circle is inscribed in the rectangle. To find the shaded portion, subtract he area of the circle from the are of the rectangle.

Area of the rectangle                                  =  11 x 12

                                                                     =  132cm²

Area of the circle with radius of 1cm          =  πr²

                                                                     = 3.142 x 1²

                                                                     = 3.142cm²

Therefore , area of the shaded  region      = 132cm²  -  3.142cm²

                                                                    = 128.86cm²

Answer:128.86

Step-by-step explanation: I got it correct on khan

Answer:

-The ordered pair (-1, 5) is a solution to the first equation because it makes the first equation true.

-The ordered pair (-1, 5) is a solution to the second equation because it makes the  second equation true.

-The ordered pair (-1, 5) is a solution to the system because it makes both equations true

Step-by-step explanation:

The correct question is

Which statements are true about the ordered pair (−1, 5) and the system of equations?

x+y=4

x−y=−6

we know that

If a ordered pair is a solution of  a equation, then the ordered pair, must satisfy the equation (makes the equation true)

If a ordered pair is a solution of the system of equations, then the ordered pair, must satisfy the equations of the system (makes both equations true)

we have

[tex]x+y=4[/tex] ----> first equation

[tex]x-y=-6[/tex] ----> second equation

step 1

Verify if the ordered pair satisfy  the first equation

Substitute the values of x and y of the ordered pair in the first equation  

For x=-1, y=5    

[tex]-1+5=4[/tex]

[tex]4=4[/tex] ---> is true

The ordered pair satisfy the first equation

so

The ordered pair is a solution to the first equation because it makes the first  equation true

step 2

Verify if the ordered pair satisfy  the second equation

Substitute the values of x and y of the ordered pair in the second equation  

For x=-1, y=5    

[tex]-1-5=-6[/tex]

[tex]-6=-6[/tex] ---> is true

The ordered pair satisfy the second equation

so

The ordered pair is a solution to the second equation because it makes the second  equation true

therefore

The ordered pair (-1, 5) is a solution to the system because it makes both equations true

Answer:

41

Step-by-step explanation:

Substitute in 5 and 7 into the correct spots:

(-3)(7)+2((8)(5)-9)

Multiply and simplify:

-21+2(40-9)

-21+2(31)

-21+62

41

:)

Answer:

50

Step-by-step explanation:

-21 + 16x - 9

-21 + 80 - 9

59 - 9

50

Answer:

F is the mid point of BC. (Proved)

Step-by-step explanation:

See the attached diagram.

Given AB ║ CD and EF is drawn to be parallel to AB.

So, EF ║ AB ║ CD .......... (1)

Now, in Δ ABD, E is the midpoint of AD and EG is parallel to AB.

So, G must be the midpoint of BD.

Now, in Δ BCD, G is the midpoint of DB and GF is parallel to CD. {From relation (1)}

So, we can write F is the midpoint of BC. (Proved)

[Since we know the theorem that if we joint the midpoints of two sides of a triangle then the line formed will be parallel to the third side.]