Solve the equation 5(x + 2) = 6x + 3x − 14, and show your work. (5 points)

Answer: (f+g)(x)=2(x-2)(x+1).

Step-by-step explanation:

[tex]f(x)=x^2-3x-4\ \ \ \ \ g(x)=x^2+x\\(f+g)(x)=(x^2-3x-4)+(x^2+x)\\(f+g)(x)=x^2-3x-4+x^2+x\\(f+g)(x)=2x^2-2x-4\\(f+g)(x)=2*(x^2-x-2)\\(f+g)(x)=2*(x^2-2x+x-2)\\(f+g)(x)=2*(x*(x-2)+(x-2))\\(f+g)(x)=2*(x-2)*(x+1).[/tex]

The average rate of change of the function over −2 ≤ x ≤ 2 is 1/2

The function is given as:

[tex]h(x) = \frac 18x^3 - x^2[/tex]

The interval is given as:

−2 ≤ x ≤ 2

Calculate h(2) and h(-2)

[tex]h(2) = \frac 18 * 2^3 - (2)^2[/tex]

h(2) = -3

[tex]h(-2) = \frac 18 * (-2)^3 - (-2)^2[/tex]

h(-2) = -5

The average rate of change is then calculated as:

[tex]m = \frac{h(-2) - h(2)}{-2-2}[/tex]

This gives

[tex]m = \frac{-5 + 3}{-4}[/tex]

Evaluate

m = 1/2

Hence, the average rate of change of the function over −2 ≤ x ≤ 2 is 1/2

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Complete question

[tex]h(x) = \frac 18x^3 - x^2[/tex]

What is the average rate of change of h over the interval −2 ≤ x ≤ 2

I'll focus on part (b) only.

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

Explanation:

Replace the variable b with the commonly used variable x and c with some other constant, let's say the letter m. These replacements will become apparent when we use the quadratic formula shortly later on.

After the replacements are made, we have -3c²-4cb+4b²​ turn into -3m²-4mx+4x²​

This is the same as 4x²​-4mx-3m²

Compare this to the format of ax²​+bx+c

We can see that

Once again, m is constant.

Let's compute the discriminant.

[tex]d = b^2 - 4ac\\\\d = (-4m)^2-4(4)(-3m^2)\\\\d = 16m^2+48m^2\\\\d = 64m^2\\\\[/tex]

This discriminant is under the square root as part of the quadratic formula.

So 4x²​-4mx-3m² would have roots of...

[tex]x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x = \frac{-b\pm\sqrt{d}}{2a}\\\\x = \frac{-(-4m)\pm\sqrt{64m^2}}{2(4)}\\\\x = \frac{4m\pm\sqrt{(8m)^2}}{8}\\\\x = \frac{4m\pm 8m}{8}\\\\x = \frac{4(m\pm 2m)}{4*2}\\\\x = \frac{m\pm 2m}{2}\\\\x = \frac{m+ 2m}{2} \ \text{ or } \ x = \frac{m-2m}{2}\\\\x = \frac{3m}{2} \ \text{ or } \ x = \frac{-m}{2}\\\\[/tex]

The whole time, the term "m" is treated as a constant. It is fixed to some unchanging number.

So why did we go through all that trouble to use the quadratic formula in the first place? It turns out that this formula is helpful to factoring quadratics. Recall that the roots of a polynomial tie very closely to its factorization.

For example, x²-4 factors to (x-2)(x+2) to have roots of x = 2 or x = -2. You could use the quadratic formula to solve x²-4 = 0 or x²+0x-4 = 0 and you should get x = 2 or x = -2 as the two solutions.

Anyways, back to the problem at hand.

Let's use the first root we found to form one of the factors needed.

The idea is to do a bit of algebra to get everything to one side. If fractions are present, then multiply both sides by the LCD to clear them out. We need to have 0 isolated on its own side.

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

So (2x-3m) is one factor

Through similar steps, you would find that the root [tex]x = \frac{-m}{2}[/tex] yields the factor of (2x+m)

Therefore, the factors are (2x-3m) and (2x+m)

------------------------------------

In summary so far, 4x²​-4mx-3m² factors to (2x-3m)(2x+m)

From here we replace x with b and replace m with c, so that we return back to the original variables used.

(2x-3m)(2x+m) becomes (2b-3c)(2b+c)

This can be confirmed using WolframAlpha. You could use the FOIL rule to expand out (2b-3c)(2b+c) and you should get 4b²​-4bc-3c²​ aka -3c²-4cb+4b²​

-------------------------------------

Those previous sections talked about part (i)

Part (ii) is where we plug in b = 4 and c = 3. In other words, we replace every copy of b with 4, and every copy of c with 3.

If you use the original expression, then,

-3c²-4cb+4b²​ = -3(3)²-4(3)(4)+4(4)²​ = -3(9)-4(12)+4(16) = -11

Or you could use the factored expression.

(2b-3c)(2b+c) = (2*4-3*3)(2*4+3) = (8-9)(8+3) = (-1)*(11) = -11

This very partially confirms that -3c²-4cb+4b² = (2b-3c)(2b+c) is indeed the case. Unfortunately you'd have to use infinitely many numeric examples to confirm the answer to part (i); however, something like the FOIL rule is far more efficient.

Function Composition exists when two functions f(x) and g(x) result in another function h(x), such that h(x) = f(g(x)). For function composition h(x) =  f(g(x)), the value of a exist -4.

Function Composition exists when two functions f(x) and g(x) result in another function h(x), such that h(x) = f(g(x)). In other words, put the outcome of one function into the other one.

Here, h(x) = f(g(x))

which means, h(x) = f(x³+ a)

[tex]$h(x) = \sqrt{x^3+a+2}[/tex]

[tex]$\sqrt{x^3-2} = \sqrt{x^3+a+2}[/tex]

To estimate the value of a, we separate equal terms:

1) Both are squared, so we can "eliminate" the square;

2) x³ = x³

3) -2 = a+2

a = -4

For function composition h(x) =  f(g(x)), the value of a exist -4.

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Answer: [tex]\Large\boxed{2}[/tex]

Concept:

Here, we need to know the idea of the PEMDAS method.

PEMDAS is an acronym for the order of operations.

Solve:

Given expression

[tex]\dfrac{3^3~-~13}{11~-~2^2}[/tex]

Simplify the exponents

[tex]=\dfrac{27~-~13}{11~-~4}[/tex]

Simplify by subtraction

[tex]=\dfrac{14}{7}[/tex]

Simplify the fraction

[tex]\Large\boxed{=2}[/tex]

Hope this helps!! :)

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Applying the angle of intersecting chords theorem, the value of z is: D. 100.

According to the angle of intersecting chords theorem, in a circle like the one shown in the image above, where two chords intersect to form vertical angles, the theorem states that the angle formed equals half of the sum of the measures of the arcs intercepted by the angles.

Applying the angle of intersecting chords theorem, let's find x first:

73 = 1/2(96 + x)

Multiply both sides by 2

2 × (73) = 1/2(96 + x) × 2

146 = 96 + x

Subtract both sides by 96

146 - 96 = 96 + x - 96

50 = x

x = 50°

Therefore:

x + z + 96 + 114 = 360° [full circle measure]

Plug in the value of x

50 + z + 96 + 114 = 360

z + 260 = 360

Subtract both sides by 260

z + 260 - 260 = 360 - 260

z = 100°

The answer is: D. 100°.

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Using the Laplace transform to solve the given integral equation  f(t) = t, 0 ≤ t < 4 0, t ≥ 4 is [tex]\frac{4(1-2e^-5s)/}{s}[/tex]

explanation is given in the image below:

Laplace remodel is an crucial remodel approach that's particularly beneficial in fixing linear regular equations. It unearths very wide programs in var- areas of physics, electrical engineering, manipulate, optics, mathematics and sign processing.

The Laplace transform technique, the function inside the time domain is transformed to a Laplace feature within the frequency area. This Laplace function could be inside the shape of an algebraic equation and it may be solved without difficulty.

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Answer:

The exact amount of time would be 8 2/9 weeks, but if we are only counting full weeks, it would be finished in 9 weeks.

Step-by-step explanation:

y = xm

18 1/2 = 2 1/4x  Change both numbers into improper fractions

18 1/2 would be 37/2  The denominator (bottom number) stays 2.  To get the top number you would (2 x 18) + 1 or 37.  That makes 37/2 equivalent to 18 1/2.

Now do the same thing to 2 1/4.  The bottom number stays the same.  We get the top number (4 x 2) + 1 or 9.  That makes 9/4 equivalent to 2 1/4.

We know have

37/2 = 9/4 x  We can solve for x by multiplying both sides of the equation by 4/9

37/2(4/9) = (9/4)(4/9)x

148/18 = x  If we divide both the top and the bottom by 2 we would get 74/9 = x  If we divide 74 by 9 we get 8 2/9

Answer:

First:

5x + 9 < 2

x= 9/5= 1,8 < 2

Second:

x + 6 < 12

x= 12/6= 2 < 12

Answer:

1

Step-by-step explanation:

f(x) = x² + 3x

g(x) = 5x + 2

f(3) = 3² + 3 x 3

f(3) = 9 + 9

f(3) = 18

g(3) = 5(3) + 2

g(3) = 15 + 2

g(3) = 17

(f - g)(3) = 18 - 17

(f - g)(3) = 1.

Answer:

Given:

Mass of the tennis ball=56.5 g

wavelength=5400 Angstroms

De-Broglie Wavelength = 5400 Angstrom.

= 5400 × 10⁻¹⁰ m.

Now, Using the Formula,

λ = h/mv

here,

h is the Planck's constant = 6.62 × 10⁻³⁴ J-s.

and v is the velocity of the tennis balls.

∴ 5400 × 10⁻¹⁰ = (6.62 × 10⁻³⁴)/ (0.054 × v)

⇒ 0.054v = (6.62 × 10⁻³⁴)/ (5400 × 10⁻¹⁰)

⇒ 0.054v = 0.0012 × 10⁻²⁴

⇒ v = 0.0012 × 10⁻²⁴/0.054

⇒ v = 0.023 × 10⁻²⁴ m/s.

Hence, the velocity of the tennis balls is 0.023 × 10⁻²⁴ m/s.

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Answer:

1/8

Step-by-step explanation:

say the bar has 20 pieces. he has 10 pieces. and he gave away 1/4 of those 10

1/4 x 10 = 2.5

20 / 2.5 = 8

So his friend recieved 1/8

you could also just do 1/2 x 1/4 which is 1/8

Answer:

1/8 of the candy bar.

Step-by-step explanation:

Elias has 1/2 of a candy bar.

He gave 1/4 of 1/2 to his friend.

[tex]\sf Friend's s\ share = \dfrac{1}{4} \ of \ \dfrac{1}{2}[/tex]

                      [tex]\sf =\dfrac{1}{4}*\dfrac{1}{2}\\\\=\dfrac{1}{8}[/tex]

Answer:

2

Step-by-step explanation:

Your slope is your change in y over your change in x.  The ordered pair is in the form of (x,y), so you subtract the y's and put that in your numerator (top)and subtract your x's and put that in your denominators (bottom).

2 - (-4) would be your top number.  To subtract a negative number is the same as adding a positive, so 2 - (-4) is the same as 2 +4, so your top number is 6.  

Now, to find the bottom number, you subtract the x's.

5-2 which is 3.

We now have the top and the bottom number

6/3 which is the same as 2.

Answer:

[tex]x=2.5+\frac{\sqrt{29}}{2}\\\\x=2.5-\frac{\sqrt{29}}{2}[/tex]

Step-by-step explanation:

So the question here is asking you to use the quadratic formula which is expressed as: [tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\[/tex]

A quadratic can generally be expressed as: [tex]y=ax^2+bx+c[/tex]

So using the equation you gave: [tex]0=x^2-5x-1[/tex]

We can identify the following values: a=1, b=-5, c=-1

Btw the equation explicitly write "1" as the coefficient of x, but since it's not provided it's implied that it's 1.

So plugging in the known values, we get the following equation:

[tex]x=\frac{5\pm\sqrt{(-5)^2-4(1)(-1)}}{2(1)}\\\\x=\frac{5\pm\sqrt{25+4}}{2}\\\\x=\frac{5\pm\sqrt{29}}{2}\\\\x=2.5+\frac{\sqrt{29}}{2}\\\\x=2.5-\frac{\sqrt{29}}{2}[/tex]

The last step just consists of taking the + and - solution, and since it asks for exact solutions you leave the 29 under the radical, and you don't approximate. There is no further simplification that can be done here.

Answer:

15?

Step-by-step explanation:

2×15=30 i am not fully sure but this should be it

Answer: (-1, 1)

Step-by-step explanation:

Since the graph is already given, we could directly refer to the intersecting point of the system of equations.

According to the graph, [ y = 2x + 3 ] and [ y = -x ] intersect at (-1, 1).

Therefore the solution is [tex]\Large\boxed{(-1,1)}[/tex]

Given equation

1) y = 2x + 3

2) y = -x

Substitute the y value in the 1) equation by the 2)

(-x) = 2x + 3

Add x on both sides

-x + x = 2x + 3 + x

0 = 3x + 3

Subtract 3 on both sides

0 - 3 = 3x + 3 - 3

-3 = 3x

Divide 3 on both sides

-3 / 3 = 3x / 3

x = -1

Substitute the x value into one of the equations to find the y value

y = -x

y = - (-1)

y = 1

Therefore, the solution is [tex]\Large\boxed{(-1,1)}[/tex]

Hope this helps!! :)

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Answer:

Associative property of addition.

Step-by-step explanation:

The location of the parentheses makes no difference to the sum of these numbers.

The property shown in the statement "(46 + 17) + 3 = 46 + (17 + 3)" is the associative property of addition.

The associative property of addition states that when adding three or more numbers, the grouping of the numbers does not affect the sum

In the given statement, we have three numbers being added: 46, 17, and 3. The statement shows that the addition is grouped differently on each side of the equation, but the result is the same.

On the left side, we have (46 + 17) + 3.

This means we first add 46 and 17, which equals 63. Then we add 3 to the sum, resulting in 66.

On the right side, we have 46 + (17 + 3). Here, we first add 17 and 3, which equals 20. Then we add 46 to the sum, also resulting in 66.

As we can see, no matter how we group the numbers being added, the final sum is the same. This demonstrates the associative property of addition.

Therefore, the property shown in the given statement is the associative property of addition.

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When we make inferences about the difference of two independent population proportions, we assume that it is a random sample, and the number of successes and failures are at least 15 in each group.

Two independent proportions tests involve comparing the proportions of two unrelated datasets.

For these two datasets to be regarded as an independent population, the following must be true or assumed to be true

Hence, the above highlights are the assumptions of two independent population proportions.

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The behavior sees the leading coefficient is already positive, then increasing the size of the inputs will simply result in the leading term being even more positively skewed. The line on the graph will start to slope to the right.

The behavior of the graph of the polynomial function f(x) as the variable x approaches either positive or negative infinity represents the end behavior of the function. The final behavior of a polynomial function's graph is determined by the degree of the function as well as the leading coefficient.

Generally, the equation for is zeros  mathematically given as

y=x^4−2x^3−11x^2+12x+36

Therefore

(x+2)^2(x-3)^2=0

x=-2

x=3

since the graph attached has a positive leading coefficient and a positive degree

In conclusion, If the leading coefficient is already positive, then increasing the size of the inputs will simply result in the leading term being even more positively skewed. The line on the graph will start to slope to the right.

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Answer:

x = 35.

Step-by-step explanation:

since they are odd, the second number isn't x + 1, it is x + 2.

x + x + 2 = 72

2x + 2 = 72

2x = 70

x = 35

To conduct a test of hypothesis with a small sample, we make an assumption that the population is normally distributed .

What is normal distribution?

A probability distribution that is symmetric about the mean is the normal distribution, sometimes referred to as the Gaussian distribution. It demonstrates that data that are close to the mean occur more frequently than data that are far from the mean.

The normal distribution appears as a "bell curve" on a graph.

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[tex]y = 2x {}^{2} - 4x - 16 \\ \delta = b {}^{2} - 4ac \\ \delta = ( - 4) {}^{2} - 4(2)( - 16) \\ \delta = 16 + 128 \\ \delta = 144 > 0 [/tex]

[tex]x = \frac{ - b + \sqrt{ \delta} }{2a} = \frac{4 + 12}{2 \times 2} = \frac{16}{4} = 4[/tex]

[tex]x = \frac{ - b - \sqrt{ \delta} }{2a} = \frac{4 - 12}{2 \times 2} \frac{ - 8}{4} = - 2[/tex]

[tex]x = 4 \: \: \: \: \: or \: \: \: \: x = - 2 \\ (x - 4) = 0 \: \: \: \: \: or \: \: \: \: \: (x + 2) = 0[/tex]

[tex](\stackrel{x_1}{-2}~,~\stackrel{y_1}{-5})\hspace{10em} \stackrel{slope}{m} ~=~ 6 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-5)}=\stackrel{m}{6}(x-\stackrel{x_1}{(-2)})\implies y+5=6(x+2)[/tex]

Concluding, there are (2 + 12) boxes of pencils left.

We know that Ms. Bloom had (4 + 7/12) boxes of pencils, and we also know that (2 + 1/12) of the boxes were broken.

The number of boxes of pencils left is given by the difference between the two above numbers, then we get:

N = (4 + 7/12) - (2 + 1/12) = (4 - 2) + (7/12 - 1/12) = 2 + 6/12

Now we can simplify the fraction:

N = 2 + 1/2

Concluding, there are (2 + 12) boxes of pencils left.

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The statement that justifies the relationship is: D. it represents a function because no length has more than one price.

For a relation to be considered a function, any of the following criteria must be met:

In the scenario given:

Length of ribbon = input (x-value)

Price of ribbon = output (y-value).

We can observe that each length of ribbon (input) has its own price (output), therefore, we can conclude that the price of ribbon is a function of the length of the ribbon.

The answer is: D. This relationship represents a function because no length has more than one price.

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Answer:

(-4,0) , (0,9)

Step-by-step explanation:

x-intercept is the point where the line touches the x-axis, which means, y = 0.

So when y = 0,

-9x + 4(0) = 36

-9x = 36

x = [tex]\frac{36}{-9}[/tex]

x = -4

Therefore, the coordinates of the x - intercept is (-4,0)

y-intercept is the point where the line touches the y-axis, which means , x = 0.

so when x = 0,

-9(0) + 4y = 36

4y = 36

y = [tex]\frac{36}{4}[/tex]

y = 9

Therefore, the coordinates of the y - intercept is (0,9)

Answer:

x - intercept (-4,0)

y -intercept (0,9)

Step-by-step explanation:

Put in 0 for x.

-9(0) + 4y = 36

4y = 36  Divide both sides by 4

y = 9

(0,9) This is where the line will cross the y axis.

Put in 0 for y

-9y + 4(0) = 36

-9y = 36 Divide both sides by -9

Y = -4