expand (x^2-3/2)^9 using binomial process

To solve for the positive number when the sum of 5 and three times the number is subtracted from the square of the number to yield zero, set up and solve the quadratic equation x^2 - 3x - 5 = 0 using the quadratic formula.

The student has asked for help in finding a positive number when the sum of 5 and three times the number is subtracted from the square of the number, and the result is 0. This involves setting up a quadratic equation and solving for the number.

Let's denote this positive number as 'x'. According to the problem statement, the equation can be written as:

x2 - (5 + 3x) = 0.

We can then expand and simplify this equation:

x2 - 5 - 3x = 0

And, rearrange it into standard quadratic form:

x2 - 3x - 5 = 0

Next, we can either factorize this equation, if possible, or use the quadratic formula to find the value of 'x' that satisfies the equation. Since this equation does not factor neatly, we use the quadratic formula:

x = ∛2 - 4ac) / 2a

In this case, a = 1, b = -3, and c = -5. Plugging these into the quadratic formula, we find the two possible solutions for x. Only the positive solution is relevant since the problem states that the number is positive.

The solutions of the given quadratic equations is 5/3 and -2

Given the quadratic function expressed as:

Factorizing the function we will have:

6x^2 + 2x – 20 = 0

3x^2 + x - 10 = 0

3x^2 +6x - 5x - 10 = 0

3x(x+2)  5(x+2) = 0

Group the function;

(3x-5)(x+2) = 0

Find the factors

3x- 5 = 0 and x + 2 = 0

x = 5/3 and -2

The solutions of the given quadratic equations is 5/3 and -2

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We have been given that she wants to spend exactly $15 on candy.

Also, she spent $9 on lollipops. It means she spent [tex]15-9= $6[/tex]  on gummy bears.

Also, we have been given that the cost of gummy bears are $0.75 per pound.

Thus, in 1$ she can purchase [tex]\frac{0.75}{6}[/tex] pounds of gummy bears.

Hence, in $6, she can purchase

[tex]\frac{1}{0.75} \times 6\\ \\ =8[/tex]

Therefore, she purchased 8 pounds of gummy bears.

Answer:92

Step-by-step explanation:

92

The tangent or tanθ in a right angle triangle is the ratio of its perpendicular to its base. The height of the building from the ground is 426.85 meters.

The tangent or tanθ in a right angle triangle is the ratio of its perpendicular to its base. it is given as,

[tex]\rm Tangent(\theta) = \dfrac{Perpendicular}{Base}[/tex]

where,

θ is the angle,

Perpendicular is the side of the triangle opposite to the angle θ,

The base is the adjacent smaller side of the angle θ.

As it is given that the angle of elevation from the person's view is 80°, while the distance between the person and the base of the building is 75 meters, therefore, using the trigonometric function the height of the building top from the person eye level is,

[tex]\rm Tangent(\theta) = \dfrac{Perpendicular}{Base}\\\\\\Tan(80^o) = \dfrac{\text{Height of the building}}{75\ meters}\\\\\\\text{Height of the building} = 75 \times tan(80^o) = 425.35\ meters[/tex]

Now, we know the height of the building top from a person's eye level, and we also know the person's eye level as well, therefore, the height of the building can be written as,

[tex]\text{Height of the building} = 425.35 + 1.5 = 426.85\rm\ meters[/tex]

Hence, the height of the building from the ground is 426.85 meters.

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The coordinates of the image after dilation with a scale factor of 2 are A'(-10, -14), B'(12, -6), C'(4, 14).

Your question involves geometry and specifically, dilation transformations. A dilation is a transformation that moves each point along the ray from the center of dilation to the point. The number given as the scale factor tells you how much to move. In this case, we are asked to dilate the given points with a scale factor of 2 centered at the origin (0,0).

So, each coordinate of the original triangle should be multiplied by 2. Let's find each coordinate after dilation:

So, the coordinates of the image after a dilation with a scale factor of 2 are A'(-10, -14), B'(12, -6), and C'(4, 14).

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To find the new balance on January 1, we need to calculate the total balance after considering purchases, returns, payments, and finance charges.

Calculate the total charges:

         Total charges = Previous balance + New purchases − Returns

         Total charges = $205.84 + $98.42 − $18.63

         Total charges = $285.63

After the payment, ascertain the extra amount due:

         Unpaid balance = Total charges − Payment

         Unpaid balance = $285.63 − $25

         Unpaid balance = $260.63

Calculate the finance charges:

         Finance charges=Unpaid balance×Monthly interest rate

         Monthly interest rate = 1.5% = 0.015

         Finance charges = $260.63 × 0.015

         Finance charges ≈ $3.91

Calculate the new balance:

         New balance = Unpaid balance + Finance charges

         New balance = $260.63 + $3.91

         New balance = $264.54

So, the new balance on January 1 is approximately $264.54.

To calculate Brent Tyson's new balance on January 1, we first determine the total charges by adding new purchases and subtracting returns from the previous balance.

Then, we deduct his payment to find the unpaid balance.

Next, we calculate finance charges at a rate of 1.5% per month on the unpaid balance.

Finally, we add the finance charges to the unpaid balance to get the new balance.

After these steps, Brent Tyson's new balance on January 1 is $264.54.

This process ensures we account for all transactions and interest charges to accurately determine the updated balance on his Discover card.

Complete Question:

The balance on Brent Tyson’s Discover card on December 1 is $205.84.

In December, he charges an additional $98.42 of purchases, has returns of $18.63, and makes a payment of $25.

If the finance charges are calculated at 1.5% per month on the unpaid balance, find the new balance on January 1.

Since Adam's expenses exceed his earnings, he will have a negative net earnings. Therefore, Adam will not earn any money this season. In fact, he will have a net loss of $7.50.

To calculate how much Adam will earn this season,  subtract his expenses from his total earnings.

Earnings per mowing session: $20

Number of mowing sessions in the grass-growing season: 18 weeks / 2 weeks = 9 sessions

Total earnings: $20 per session * 9 sessions = $180

Expenses:

Cost of the lawn mower: $165

Cost of gasoline per session: $2.50 per session * 9 sessions = $22.50

Total expenses: $165 + $22.50 = $187.50

To find Adam's net earnings, subtract his expenses from his total earnings:

Net earnings = Total earnings - Total expenses

Net earnings = $180 - $187.50

Net earnings = -$7.50

Since Adam's expenses exceed his earnings, he will have a negative net earnings. Therefore, Adam will not earn any money this season. In fact, he will have a net loss of $7.50.

Adam charges $20 to mow his neighbors lawn every 2 weeks for the 18 weeks of grass-growing season. He wants to buy a nice lawn mower for $165 and gasoline costs $2.50 every 2 weeks.  How much will Adam earn this season?

Answer:

140 m

Step-by-step explanation:

A kite is a quadrilateral with two distinct pairs of adjacent sides that are equal in length.

The perimeter of the kite is the sum of the length of all its sides. So, the perimeter of kite ABCD can be expressed as:

[tex]\sf Perimeter = AB + BC + CD + DA[/tex]

As BC = AB and CD = DA, then:

[tex]\sf Perimeter = BC + BC + CD + CD\\\\Perimeter = 2(BC + CD)[/tex]

Therefore, to find the perimeter of kite ABCD, we need to determine the lengths of sides BC and CD.

The diagonals of a kite intersect to form two pairs of congruent right triangles, where the sides of the kite are the hypotenuses of these triangles.

in kite ABCD:

The longer diagonal of a kite always bisects the shorter diagonal at a right angle.

Let M be the point of intersection of the diagonals. Therefore:

AM = MC = 24 m

Side BC is the hypotenuse of right triangle BMC, with legs measuring 18 m and 24 m, while side CD is the hypotenuse of right triangle CMD, with legs measuring 32 m and 24 m.

To find the lengths of sides BC and CD, we can use the Pythagorean Theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two legs.

[tex]\sf BC^2=18^2+24^2 \\\\BC^2=324+576 \\\\BC^2=900\\\\BC=\sqrt{900}\\\\BC=30\; m[/tex]

[tex]\sf CD^2=32^2+24^2 \\\\CD^2=1024+576 \\\\CD^2=1600\\\\CD=\sqrt{1600}\\\\CD=40\; m[/tex]

Now we have determined the length of sides BC and CD, we can substitute them into the perimeter formula:

[tex]\sf Perimeter = 2(30+40) \\\\ Perimeter = 2(70) \\\\ Perimeter = 140 \; m[/tex]

Therefore, the perimeter of kite ABCD is:

[tex]\LARGE\boxed{\boxed{\sf 140\; m}}[/tex]

Answer:

C)square 37 - square root 26

Step-by-step explanation:

The solution to the system of equations is [tex]\(x = -3\)[/tex] and [tex]\(y = 2\).[/tex]

let's solve the system of equations using the substitution method.

Given:

1. [tex]\(3x + 2y = -5\)[/tex]

2.[tex]\(2x - 5y = -16\)[/tex]

Step 1: Solve one equation for one variable in terms of the other variable.

Let's solve equation 1 for [tex]\(x\):[/tex]

[tex]\[3x = -2y - 5\][/tex]

[tex]\[x = \frac{-2y - 5}{3}\][/tex]

Step 2: Substitute the expression for [tex]\(x\)[/tex]from step 1 into the other equation.

Substitute [tex]\(x = \frac{-2y - 5}{3}\)[/tex] into equation 2:

[tex]\[2\left(\frac{-2y - 5}{3}\right) - 5y = -16\][/tex]

Step 3: Solve for [tex]\(y\).[/tex]

[tex]\[ \frac{-4y - 10}{3} - 5y = -16 \][/tex]

[tex]\[ -4y - 10 - 15y = -48 \][/tex]

[tex]\[ -19y - 10 = -48 \][/tex]

[tex]\[ -19y = -38 \][/tex]

[tex]\[ y = 2 \][/tex]

Step 4: Substitute the value of [tex]\(y\)[/tex] into one of the original equations to solve for [tex]\(x\).[/tex]

Let's use equation 1:

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

[tex]\[3x + 4 = -5\][/tex]

[tex]\[3x = -9\][/tex]

[tex]\[x = -3\][/tex]

So, the solution to the system of equations is [tex]\(x = -3\)[/tex] and [tex]\(y = 2\).[/tex]

Complete question:

solve by substitution 3x+2y=-5, 2x-5y=-16

By continuing the pattern of Amy increasing her volunteer hours by 2 each month, she will work for 30 hours in December.

To find out how many hours Amy will work in December, we must first identify the number of terms between June and December.

Counting inclusively, we see that there are seven months between June and December (June, July, August, September, October, November, December), which means we should look for the 7th term after June to find the number of hours Amy will work in December.

Since we have established an increase of 2 hours each month, we can calculate this by adding 7 terms of an arithmetic progression to Amy's last known volunteering hours.

To do this, we use the explicit formula for an arithmetic sequence:

Tn = a + (n - 1) * d,

where Tn is the term we wish to find, a is the first term in the sequence, n is the position of the term in the sequence, and d is the common difference between terms. Amy's last known volunteering hours in June (the 4th term in the sequence) were 16, with a common difference of 2.

Adding 7 intervals of 2 hours each gives us 14 additional hours by December.

Therefore, we add this to the 16 hours she worked in June: 16 + 14 = 30 hours.

Amy is expected to volunteer for 30 hours in December.

Final answer:

The domain of the function h(x)=9x/x(x²-36) is all real numbers except 0, 6, and -6, because the function is undefined at these points.

In conclusion, the domain of h(x) is x ∈ ℝ, x ≠ 0, x ≠ 6, x ≠ -6.

Explanation:

The question asks to determine the domain of the function h(x) = 9x / x(x² - 36).

The domain of a function consists of all the real numbers x for which the function is defined. In this case, the function will be undefined when the denominator is zero because division by zero is not defined in mathematics.

To find these values, we set the denominator equal to zero and solve the resulting equation:

This can be factored further as x(x - 6)(x + 6) = 0, which gives us the zero points of x = 0, x = 6, and x = -6.

Therefore, the domain of h(x) is all real numbers except 0, 6, and -6.

In conclusion, the domain of h(x) is x ∈ ℝ, x ≠ 0, x ≠ 6, x ≠ -6.

Answer:

81

Step-by-step explanation:

To find the points of discontinuity and sketch the graph of the rational function (x^2-4)/(x+2)(x^2-49), we need to determine the values of x that make the denominator equal to zero. We find that x = -2, x = 7, and x = -7 are the points of discontinuity. We can then sketch the graph by plotting the vertical asymptotes at x = -2, x = 7, and x = -7, and determining the behavior of the function on each side of those points.

To find the points of discontinuity for the rational function (x^2-4)/(x+2)(x^2-49), we need to determine where the denominator is equal to zero. The denominator has two factors, (x+2) and (x^2-49). Setting each factor equal to zero, we find that x = -2, x = 7, and x = -7. These are the points where the function has vertical asymptotes or holes.

To sketch the graph, we can start by plotting the vertical asymptotes at x = -2, x = 7, and x = -7. Next, we can examine the sign of the function on each side of the vertical asymptotes to determine the behavior of the graph. Finally, we can plot any holes in the graph at the corresponding x-values.

Additionally, we should determine if there are any horizontal asymptotes. For rational functions, horizontal asymptotes occur when the degree of the numerator is less than or equal to the degree of the denominator. In this case, the degree of the numerator is 2 and the degree of the denominator is 3, so there is no horizontal asymptote.

Answer:

The graph for 12x-9y=36 is given below

Step-by-step explanation:

The graph for 12x-9y=36 is given below.

[tex]12x-9y=36\\\frac{12x}{36}+\frac{9y}{-36}=1\\\frac{x}{3}+\frac{y}{-4}=1[/tex]

So x intercept is 3 and y intercept is -4.

The required graph is shown below which represents the equation 12x - 9y = 36.

To graph the equation 12x-9y=36, we can rearrange it to slope-intercept form, which is y=mx+b, where m is the slope and b is the y-intercept.

First, solve for y by subtracting 12x from both sides and then dividing by -9. This gives us y=-4/3x+4.

Now we can see that the slope is -4/3 and the y-intercept is 4.

To graph this line, we can start at the y-intercept of (0,4) and then use the slope to find another point.

Since the slope is negative, we can move down 4 units (which is the denominator of the slope) and then move to the right 3 units (which is the numerator of the slope) to get another point at (3,0).

We can then draw a straight line through these two points to graph the equation.

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