simplify the expression

The nature of roots of the quadratic equation ax²+bx+c= 0 can be describe in three ways depending on the value of b²-4ac.

For quadratic equation ax²+bx+c=0 if

Case 1) If b²-4ac>0  then the roots of equation will be real and distinct.

Case 2) If b²-4ac=0  then roots will be real and equal.

Case 3) If b²-4ac<0 then roots will be imaginary and complex number pairs.

also a) if b2-4ac >0 and perfect square then the roots are real,distinct, and rational

b) if b2-4ac>0 and not a perfect square then the roots are real, distinct but irrational.

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The maximum value of f(x,y) on the unit of circle is [tex]\frac{48 \sqrt{21}}{2,401}[/tex].

The full question is in the attachment. From the unit circle

Substitute to f(x,y)

f(x,y) = x³y⁴ = x³ (y²)² = x³ (1 - x²)²

f(x,y) = x³ (1 - 2x² + x⁴)

f(x,y) = x³ - 2x⁵ + x⁷

df/dx = 3x² - (2 × 5)x⁴ + 7x⁶

df/dx = 3x² - 10x⁴ + 7x⁶

df/dx = x² (3 - 10x² + 7x⁴)

To find the maximum or minimum value, the derivated equals zero.

0 = x² (7x⁴ - 10x² + 3)

0 = x² (x - 1) (x + 1) (7x² - 3)

7x² - 3 = 0

Using ABC formula

[tex]x_{1,2} \:=\: \frac{-b \pm \sqrt{b^2 \:-\: 4ac}}{2a}[/tex]

[tex]x_{1,2} \:=\: \frac{-0 \pm \sqrt{0^2 \:-\: (4 \times 7 \times - 3)}}{2 \times 7}[/tex]

[tex]x_{1,2} \:=\: \frac{\pm \sqrt{4 \times 21)}}{14}[/tex]

[tex]x_{1,2} \:=\: \frac{\pm 2 \sqrt{21}}{14}[/tex]

[tex]x_{1,2} \:=\: \frac{\pm \sqrt{21}}{7}[/tex]

The maximum value from x₁.

The minimum value from x₂.

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Answer:

3510 marbles

Step-by-step explanation:

45 x 6 = 270

270 x 13 = 3510

Hope this helps :)

Answer:3510

Step-by-step explanation:

:)

Answer:

to move te function up z units, add zto the whole equation

from

y=sqrt(x) to

y=sqrt(x)+2

parenth function moved up 2 units

my hands are cold so I don't type well

Step-by-step explanation:

The value of x when the quadratic expression  t² - 8t + 16 is factorised to (t + a)² is -4.

Quadratic expression is an expression with the variable with the highest power of 2. The quadratic equation in its standard form is ax² + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term.

Therefore, let's factorised the expression t² - 8t + 16 in the form (t + a)². Where a is an integer.

Hence,

t² - 8t + 16

t² - 4t - 4t + 16

t(t - 4) -4(t - 4)

(t - 4)(t - 4)

Hence,

(t + a)² = (t - 4)²

Therefore,

a = -4

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According to the given statement the area of the cross section of the cone is 72 square units.

a solid produced by rotating one of the legs of a right triangle. A solid that is surrounded by a circle and other closed planar base and indeed the surface created by line segments connecting each point here on base's perimeter to a single vertex is known as a right circle cone; for more information, see the Density Formulas Table.

From the given information the height is 16 inches and the radius is 4.5 inches becuase the cone is sliced by a plane that passes through the center of the base, and also passes through the top vertex

So, The radius would be half the base, so the base would be 4.5 * 2 = 9 inches.

The area of the cross section of the cone is found by multiplying the base by the height, then dividing by two.

Area of the cross section of the cone = (9 x 16)/2 = 144

144 / 2 = 72 square inches.

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Answer:

PA = PB

Step-by-step explanation:

The radius OA = OB

so triangles OPA is congruent to OPB because OA = OB, OP is the same side, angle OPA = angle OPB = 90o

therefore PA = PB

Answer:

The total cost of the materials for the pillows is $172.37

Step-by-step explanation:

First, we need to calculate how many yards of each type of fabric Kelly needs to buy. She needs 2 linen pillows, which means she needs 22 = 4 yards of linen fabric.

She also needs 2 satin pillows, which means she needs 22 = 4 yards of satin fabric.

She also needs 1 faux fur pillow, which means she needs 12 = 2 yards of faux fur fabric.

The total cost of the linen fabric is 4*$13.12 = $52.48.

The total cost of the satin fabric is 4*$9.99 = $39.96.

The total cost of the faux fur fabric is 2*$14.99 = $29.98.

The total cost of the materials for the pillows is $52.48 + $39.96 + $29.98 = $122.42.

Finally, Kelly needs to buy 5 pillow inserts at $9.99 each, which means she needs to spend 5*$9.99 = $49.95 on pillow inserts.

The total cost of the materials for the pillows is $122.42 + $49.95 = $172.37.

Answer:

x >= 1/3

Step-by-step explanation:

y = (e^x + 1)/3

x = (e^y + 1)/3   ==> inverse function is when you switch x and y

3*x = 3 * (e^y + 1)/3   ==> solve for y by first multiplying both sides by 3

3x = e^y + 1

3x - 1 = e^y + 1 - 1

3x - 1 = e^y  ==> now take natural log on each side to isolate y

ln(3x - 1) = ln(e^y)

y = ln(3x - 1)  ==> simplify

[tex]f^{-1}[/tex](x) = ln(3x - 1)  ==> plug in [tex]f^{-1}[/tex](x) for y to indicate the inverse function

3x - 1 >= 0  ==> you can't have the log function of a negative number

3x - 1 + 1 >= 0 + 1  ==> add 1 on both sides to isolate x

3x >= 1

3x/3 >= 1/3

x >= 1/3

Construct an analysis of variance table is given below in the answer.

Temperature in mathematics is a measure of how hot or cold something is compared to a reference point. Temperature is a scalar quantity, meaning it has magnitude but no direction. Temperature is measured in units such as Celsius, Fahrenheit, and Kelvin. Temperature is typically associated with thermodynamic systems in physics, which can be modeled mathematically.

Analysis of Variance Table
Source of Variation               Sum of Squares      Degrees of Freedom   Mean Square      F-test          Significance Level
Temperature                       65.6                  2                   32.8              5.14             0.05
Error                             57.4                  12                  4.78
Total                             123                   14

The F-test statistic (5.14) is greater than the critical value of 3.89, indicating that the temperature level has a significant effect on the mean yield of the process at the 0.05 level of significance.
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Answer:

In the set {1, 1, 2, 2, 3, 3, 4, 4}, the value that occurs most frequently is 1, 2, 3, and 4, so the mode of this set is 1, 2, 3, and 4...

The mode of a set of numbers is the value that occurs most frequently in the set. In the set {1, 1, 2, 2, 3, 3, 4, 4}, the value that occurs most frequently is 1, 2, 3, and 4, so the mode of this set is 1, 2, 3, and 4.

It's important to note that a set of numbers can have more than one mode if multiple values occur with the highest frequency. A set of numbers that has multiple modes is called a multimodal set. In the example set {1, 1, 2, 2, 3, 3, 4, 4}, all four values occur with the same frequency, so the set is multimodal.

On the other hand, if no value occurs more frequently than any other value, the set is said to have no mode. For example, the set {1, 2, 3, 4, 5} has no mode because all values occur with the same frequency.

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The required quotient would be [tex]66\frac{8}{25}[/tex] or 66.32 which is determined by dividing 1,658 by any divisor 25.

The result of dividing an integer by any so-called quotient divisor is referred to as the quotient. The divisor appears in the dividend a certain number of times.

Arithmetic operations can also be specified by subtracting, dividing, and multiplying built-in functions. The operator that performs the arithmetic operation is called the arithmetic operator.

Multiplication operation: Multiplies values on either side of the operator

For example [tex]12*2=24[/tex]

Division operation: Divides left-hand operand by right-hand operand

For example  [tex]\frac{4}{2} =2[/tex]

Dividend is equal to 1,658

Divisor is equal to 25

Unit place value of 1658 is equal to 8.

25 is not a factor of 1658

Smallest number to 1658 divisible by 25 is 1650

Divide 1658 by 25 we get remainder

Dividend = ( Divisor × quotient) + remainder

1658 = 1650 + 8

1658 = ( 25 × 66 ) + 8

Remainder is equal to 8

Quotient is equal to 66

In decimal form

⇒ 1658 ÷ 25

Apply the Division operation, and we get

⇒ [tex]\frac{1658}{25}[/tex]

⇒ 66.32

⇒  [tex]66\frac{8}{25}[/tex]

Therefore,

The required quotient would be [tex]66\frac{8}{25}[/tex] or 66.32 which is determined by dividing 1,658 by any divisor 25.

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The end behavior of the graph of the polynomial function y = 10x^9 + 4x is asymptotic to negative infinity as x approaches negative infinity, and asymptotic to positive infinity as x approaches positive infinity.

The end behavior of a polynomial function is determined by the degree of the polynomial. The polynomial function y = 10x^9 + 4x is a degree 9 polynomial, so it will exhibit an asymptotic behavior in which the graph approaches either positive or negative infinity as x approaches either positive or negative infinity. In this case, the graph will approach negative infinity as x approaches negative infinity, and it will approach positive infinity as x approaches positive infinity. This behavior is due to the fact that the highest degree term (10x^9) will dominate the behavior of the graph as x gets very large in either the positive or negative direction. The lower degree terms (4x) will have a smaller effect on the end behavior of the graph.

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The required value zeros of the given polynomial are given as -1, -2, and -3.

Zeros of the polynomial function are the values of the dependent variable, when putting that value in the expression, the expressions explicitly give zero.

Here,
Let the given expression is equal to zero,
x³+ 6x²+ 11x+ 6 = 0

Factorization of the above expression gives.
(x + 1)(x + 2)(x+3) = 0

Now, for the zeros,
x + 1 = 0;  x +2 = 0;  x + 3 = 0
x = -1     ;  x = -2    ;  x = -3  

Thus, the zeros of the given expression's zeros are -1, -2, and -3.

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The correct answer for this is x= -1, x = -2 and x =-3

The quadratic equation in standard form given the sum and product of roots is

x² - (α + β)x + αβ = 0.

Polynomial is an equation written as the sum of terms of the form kx^n.

where k and n are positive integers.

We have,

A polynomial with degree 2 is called a quadratic equation.

Now,

The standard form of a quadratic equation is ax² + bx + c.

Now,

Let the roots of the quadratic equation be α and β.

α + β = -b/a

αβ = c/a

Thus,

The quadratic equation is x² - (α + β)x + αβ = 0.

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The length of r, to the nearest 10th of an inch, is 2.4 inches.

The relationship between the sides and angles of non-right (oblique) triangles is defined by the Law of Sines. Simply put, it states that the ratio of the length of a triangle's side to the sine of the angle opposite that side is the same for all triangle sides and angles.

Given that in ΔQRS, s = 5.2 inches, ∠S=129° and ∠Q=30°. The value of side r will be calculated by using the sin angle theorem.

The value of r is,

r / sin21 = s / sin129

r = 5.2 x ( sin21/sin129)

r = 5.2 x 0.46

r = 2.4 inch

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Answer:

A) [tex]{2s + 5t = 49.50\\\\s = 2t[/tex]

B) Keegan paid $5.50 for each t-shirt and $11 for each short.

Step-by-step explanation:

A)

⭐ What is a system of equations?

We are asked to create a system of equations for Keegan's situation.

Let s = the price for every short

Let t = the price for every t-shirt

So, if Keegan buys 2 shorts and 5 t-shirts, his cost will be $49.50.

As an equation, this situation would be:

[tex]2s + 5t = 49.50[/tex]

It's given that the cost of 1 short is twice the cost of 1 t-shirt.

As an equation, this situation would be:

[tex]s = 2t[/tex]

B)

⭐How can we solve a system of equations?

For this problem, the best way to solve this system of equations is through substitution. I will demonstrate how to use substitution for this problem.

Using the equations [tex]2s + 5t = 49.50[/tex] and [tex]s = 2t[/tex], we will literally substitute a variable from one equation into another equation to solve for the other variable.

Given that [tex]s = 2t[/tex]:

[tex]2s + 5t = 49.50[/tex]

[tex]2(2t) + 5t = 49.50[/tex]

[tex]4t + 5t = 49.50[/tex]

[tex]9t = 49.50[/tex]

[tex]t = 5.50[/tex]

∴ It costs Keegan $5.50 for one t-shirt.

Now, we can substitute the value of t into the same equation to solve for the value of s.

Given that [tex]t = 5.50[/tex]:

[tex]2s + 5(5.50) = 49.50[/tex]

[tex]2s + 27.5 = 49.50[/tex]

[tex]2s = 22[/tex]

[tex]s = 11[/tex]

∴ It costs Keegan $11 for one short.

6) The time required by James if he completes the work alone is 18 hours.

Time and work are concerned with the amount of time it takes a person or a group of people to complete a task and the effectiveness of the work performed by each of them. The fundamental idea behind Time and Work is comparable to the idea behind Proportionality, which is present in all arithmetic disciplines.

"How much work one person can complete in one day is what the term "work efficiency" refers to. For instance, someone can complete a task in two days. In other words, he may complete 50% of the work in a single day. His effectiveness will therefore be 50%.

Given that,

Monica is fencing a yard

Monica took 9 hours to fence the yard.

If she takes James help to fence the yard then she estimates that she can do it in 6 hours.

Now, we have to find how much time James requires if he work alone=?

Then,

(9x)/(9+x)=6

9x=54+6x

3x=54

x=18

So, the time required by James if he completes the work alone is 18 hours.

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The function H(x) = 3x⁴ + 2x — 5/x + 9x³ - 7 is not a polynomial function

A polynomial function is a function in an equation, such as the quadratic equation, cubic equation, etc., that only uses non-negative integer powers or only positive integer exponents of a variable.

A polynomial with an exponent of 1 is, for instance, 2x+5.

The given polynomial in standard form is written as

H(x) = 3x⁴ + 2x — 5/x + 9x³ - 7

H(x) = 3x⁴ + 9x³ + 2x  - 7 - 5x⁻¹

this shows that on of the term has a negative exponent 5x⁻¹ and hence not a polynomial

the leading coefficient is 3

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The discriminant of a 2x2 matrix is equal to the determinant of the matrix squared. The determinant is calculated by subtracting the product of the two diagonal elements from the product of the two off-diagonal elements

To find the discriminant of a 2x2 matrix, first calculate the determinant of the matrix. The discriminant is then equal to the determinant of the matrix squared.

For example, consider the 2x2 matrix:

A = [a b]

[c d]

The determinant of A is equal to (ad - bc). The discriminant of A is then equal to (ad - bc)^2.

The discriminant of a 2x2 matrix is equal to the determinant of the matrix squared. The determinant is calculated by subtracting the product of the two diagonal elements from the product of the two off-diagonal elements.

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The book should be sold at Rs.181.49 to receive a 5% profit.

In mathematics, a percentage is a number or ratio that represents a fraction of 100. It is one of the ways to represent a dimensionless relationship between two numbers; other methods include ratios, fractions, and decimals. Percentages are often denoted by the symbol "%" written after the number.

Given here, CP for Sajan = 180

and SP for Sajan = (1+20%)

                            = 1.2 × 180

                            = 216

CP for Niranjan = 216 and SP = 0.8 × 216

                                              = 172.8

CP for Niranjan = 172.8

therefore to sell it at 5% profit we get  SP for Niranjan = 172.8 × 1.05

                                                                                          = 181.49

Hence selling price for Niranjan is equal to Rs.181.49                  

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Answer:

x=72

Step-by-step explanation:

In the XY plane, the circles with equations (x-3) + y = 11 and (x+2) + y = r are externally tangent. the value of r is -6.

Generally, To find the value of r, we can start by solving for the centers and radii of the two circles.

The equation of the first circle is of the form (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. We can rewrite the equation of the first circle as (x - 3)² + y² = 11², so the center of the circle is (3, 0) and the radius is 11.

The equation of the second circle is also of the form (x - h)₂ + (y - k)₂ = r². We can rewrite the equation of the second circle as (x + 2)² + y² = r², so the center of the circle is (-2, 0) and the radius is r.

Since the two circles are externally tangent, the distance between their centers is equal to the sum of their radii. Therefore, the distance between the centers is equal to 11 + r. We can use the distance formula to find the distance between the centers:

distance = √((h₂ - h₁)^2 + (k₂ - k₁)²)

Plugging in the coordinates of the centers, we get:

distance = √((-2 - 3)^2 + (0 - 0)^2) = √(5^2 + 0^2) = √(25) = 5

Since the distance between the centers is equal to the sum of the radii, we can set up the equation:

5 = 11 + r

Solving for r, we get:

r = 5 - 11 = -6

Therefore, the value of r is -6.

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On solving the provided question, we can say that - There exist 333 integers smaller than 1000 that are evenly divisible by 3 if you divide 1000 by 3, which results in 333 with a residual of 1.

The divisibility rule is a concise and practical approach to check, often by looking at the integer's digits, whether a given integer is divisible by a given set divisor without actually executing division. Without actually doing the division procedure, you may quickly discover if a given number can be divided by a defined divisor using the divisibility test. When dividing two numbers exactly, the quotient must be an integer and the remainder must be zero.

here,

1000/3

remainder = 1

divisor = 333

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The inverse of 7 modulo 11 is 4, which means that 4 multiplied by 7 gives a number congruent to 1 (mod 11). This means that 4 is the number that when multiplied by 7 gives a remainder of 1 when divided by 11.

In modular arithmetic, the inverse of a number is the number that when multiplied by the original number gives a result that is congruent to 1 (modulus). For example, if we take the modulus of 11 as an example, then the inverse of 7 modulo 11 is 4, since 7*4 = 28 ≡ 1 (mod 11). This means that 4 is the number that when multiplied by 7 gives a remainder of 1 when divided by 11. The inverse of a number modulo a given modulus can be calculated using the Extended Euclidean algorithm. This algorithm allows us to find the inverse of a number modulo a given modulus by solving a series of equations and using the result to calculate the inverse. For example, in the case of 7 mod 11, the Extended Euclidean algorithm gives us the result 4, which is the inverse of 7 mod 11.

11 = 7*1 + 4

7 = 4*1 + 3

4 = 3*1 + 1

3 = 1*3 + 0

By reading the equations from right to left, we can see that the inverse of 7 mod 11 is 4, since 4*7 = 28 ≡ 1 (mod 11).

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The median of 5 and 7 is 6.

The median is the middle point in a dataset—half of the data points are smaller than the median and half of the data points are larger.

A data set's median value is the point where 50% of the data points have values that are lower or equal to it, and 50% of the data points have values that are higher or equal to it.

To arrange the data points in ascending order for a small data set, count the number of data points (n).

To get the rank of the data point whose value is the median when the number of data points is unequal, multiply the total by 1 and divide the results by 2.

Median = (5+7)/2 = 12/2 = 6

Thus, the median of 5 and 7 is 6.

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To get the rank of the data point whose value is the median when the number of data points is unequal, multiply the total by 1 and divide the results by 2.

Median = (5+7)/2 = 12/2 = 6

Thus, the median of 5 and 7 is 6.

Answer:

Below

Step-by-step explanation:

Replace the '8' with any number

here is one

y = 7 x + 4

To find the missing term, we can use the expansion of the square of a binomial, which is given by the formula (a + b)^2 = a^2 + 2ab + b^2. In this case, the binomial is x + 12, so we can use the formula like this:

[tex](x + 12)^2 = x^2 + 2 * x * 12 + 12^2[/tex]

Therefore, the missing term is 144.

You can also find the missing term by expanding the left-hand side of the equation using the same formula:

[tex](x + 12)^2 = (x^2 + 24x) + __[/tex]

When we expand the left-hand side using the formula, we get:

[tex](x + 12)^2 = x^2 + 24x + 144[/tex]

So the missing term is also 144.

The solution of the equation is x = -1 ± i√6.

What is solution of an equation?

An assignment of values to the unknown variables that establishes the equality in the equation is referred to as a solution. To put it another way, a solution is a value or set of values (one for each unknown) that, when used to replace the unknowns, cause the equation to equal itself.

Given equation is

x² + 2x + 7 = 0

Compare the above equation with ax² + bx + c = 0:

a = 1, b= 2, c = 7.

Now applying quadratic formula x = [-b ± √(b² - 4ac)]/2a:

x = [-2 ± √(2² - 4×1×7)]/(2×1)

x =  [-2 ± √(4 - 28)]/(2×1)

x =  [-2 ± √(-24)]/2

x =  [-2 ± 2i√6]/2

x =  -1 ± i√6

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