prove that the roots of x² + (1 - k)x + k - 3 = 0 are real for all real values of k​

Marcus jogged [tex]2\frac{1}{2}[/tex] miles over the weekend.

Marcus jogged [tex]1\frac{2}{5}[/tex] miles on Saturday and 9/10 miles on Sunday.

we first need to find a common denominator:

[tex]1\frac{2}{5}[/tex]  = 7/5

9/10 = 0.9 miles

Now, we can round each of these to the nearest 1/2:

7/5 is closer to [tex]1\frac{1}{2}[/tex] than 1, so we round to [tex]1\frac{1}{2}[/tex]

0.9 is closer to 1 than 1/2, so we round to 1

Adding these rounded values, we get:

[tex]1\frac{1}{2}[/tex] + 1 =[tex]2\frac{1}{2}[/tex]

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Answer:

Therefore, the fully factorised form is 5a(1 + 3b).

Step-by-step explanation:

5a + 15ab as 5a(1 + 3b) by taking out the common factor of 5a from both terms.

11, - 6, -1, 4, 9, 14, 19, ... are mapped onto 4. ... A;-l = (-ltQn (mod N),. (A5.34) ... 131 2, 14, 34, 38, 42, 78, 90, 178, 778, 974(1000).

Step-by-step explanation:

the nearest .1/2 incp, we say the 'unit 131/2 incl. 1.4 a measUrement is-stated tp- be 546 inch, this UM= the

If the radius is 3 cm, the height of the cylinder is approximately 7.06 cm, which is closest to option B, 7 cm. The correct option is B.

The formula for the volume of a cylinder is V = πr²h, where V is the volume, r is the radius, and h is the height.

We are given that the volume of the cylinder is 637 cm³ and the radius is 3 cm. Substituting these values into the formula, we get:

637 = π(3²)h

Simplifying:

637 = 9πh

h = 637 / (9π)

h ≈ 7.06

Thus, the height of the cylinder is approximately 7.06 cm, which is closest to option B, 7 cm.

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Answer:

a = 3

Step-by-step explanation:

Collect like-terms:

[tex] - 16 = a - 19[/tex]

[tex]a = - 16 + 19[/tex]

[tex]a = 3[/tex]

Answer: 3

Step-by-step explanation: you take 19 from 3 giving you -16

By translation, the image of the triangle X is equivalent to the triangle E.

Herein we find the representation of a triangle, whose vertices are described below:

A(x, y) = (5, 5), B(x, y) = (5, 6), C(x, y) = (7, 5)

Whose image must be determined by a rigid transformation known as translation:

P'(x, y) = P(x, y) + T(x, y)

Where:

Now we proceed to determine the images of the vertices of the triangle:

A'(x, y) = (5, 5) + (4, - 3)

A'(x, y) = (9, 2)

B'(x, y) = (5, 6) + (4, - 3)

B'(x, y) = (9, 3)

C'(x, y) = (7, 5) + (4, - 3)

C'(x, y) = (11, 2)

Therefore, the triangle E is the image of the triangle X.

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Step-by-step explanation:

well, the combination of all set elements.

FYI : if some elements appear in both sets, we only use these once for the result set (this is not the case here, but it might be in other questions).

the combination of both sets is then

{b, c, d, e, g, h}

4/5 is a real number.

A real number is a value in mathematics that expresses a quantity along a continuous line. Real numbers can be expressed as fractions or decimals and can be positive, negative, or zero.

4/5 is a rational number in this situation, which is a form of real number that may be written as a ratio of two integers. A real number can also be irrational, which means that it cannot be stated as a ratio of two integers and that its decimal representation goes on indefinitely without repeating.

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Answer:

jordans van

Step-by-step explanation:

yes satiscal

Answer: correct answer is A 2/3x^2

Step-by-step explanation:Multiply each side of the equation by 2cos2θ to get 2rcos2θ=3sinθ.

Multiply each side of the equation by r to get 2r2cos2θ=3rsinθ. Using the relationships x=rcosθ and y=rsinθ, the equation becomes 2x2=3y.

Divide each side of the equation by 3 and reverse the order of the equation to find that y=2/3x^2

Answer:

An inequality that represents this situation is:

x + 5 ≤ 11

where x is the number of outfits you can pack in addition to the 5 you already have.

Step-by-step explanation:

i got a what if If you have 6 outfits packed already, then the inequality becomes:

x + 6 ≤ 11

where x is the number of outfits you can pack in addition to the 6 you already have. happy to help

The rate of the speeding car is 0.493.

We have,

The police car is at 80 mph and it is 44 yards from the intersection, is increasing at 45 mph, while the other car is 117 yards away from the intersection.

So, Rate of speeding car

= (45- 80)/ (117- 44)

= -35/(-73)

= 35/73

= 0.493

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The values of the trigonometric functions are

1. sin A = 12/13

2. cos A = 5/13

3. Tan A = 12/5

4. cot A = 5/12

5. cosec A = 13/12

6. sec A = 13/5

The trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. Examples of these functions are; sin, cos , tan cot, cosec , sec e.t.c

Sin(tetha) = opp/hyp

cos(tetha) = adj/hyp

tan(tetha) = opp/adj

here the opposite side to angle A is 12 and the adjascent is 5 while the hypotenuse is 13.

Therefore,

1. sin A = 12/13

2. cos A = 5/13

3. Tan A = 12/5

4. cot A =1/tan A = 5/12

5. cosec A = 1/sin = 13/12

6. sec A = 1/cos = 13/5

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The area of the green region of the circle is 96.4 ft².

Area is the region bounded by a plane shape.

To calculate the area of the green region of the circle, we use the formula below

Formula:

Where:

From the diagram,

Given:

Substitute these values into equation 1

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The area of the green region in the circle given is expressed as approximately: a. 96.5 square feet.

The sector of a circle is calculated as:

A = ∅/360 * πr², where r is the radius and ∅ is the reference or central angle of the sector.

Given the following:

Central or reference angle (∅) = 180 - 27 = 153 degrees

Radius (r) = 17/2 = 8.5 feet

Plug in the values:

Area of the green region = 153/360 * π * 8.5²

Area of the green region ≈ 96.5 square feet.

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Answer:

(2, -5)

(4, -3)

(3, -1)

Step-by-step explanation:

When reflected across the X-axis, the sign of the y will change to the opposite.

(x, y) → (x, -y)

Let's solve

(2,5) → (2, -5)

(4,3) → (4, -3)

(3,1) → (3, -1)

So, the missing coordinates are: (2, -5)

                                                       (4, -3)

                                                       (3, -1)

Expression used for area of a rectangle is 12x² + 74x + 44 .

Its degree and classification is 2 and quadratic

and demonstration of a closure property is given by product of a polynomial is a polynomial.

Length of the rectangle = (6x + 4) units

and Width of the rectangle = (2x + 11) units.

The area of the rectangle is given by the product of its length and width,

Area = (6x + 4) × (2x + 11)

Find the product,

⇒ Area = 6x × 2x + 6x × 11 + 4 × 2x + 4 × 11

Simplifying this expression, we get,

⇒ Area = 12x² + 74x + 44

The expression that represents the area of the rectangle is 12x² + 74x + 44.

The degree of the expression 12x² + 74x + 44 is 2,

since the highest power of x is 2.

The classification of this expression is quadratic.

Part A demonstrates the closure property for polynomials because the product of two polynomials is also a polynomial.

Here, the area of the rectangle is the product of two polynomials (6x + 4) and (2x + 11).

And the result is also a polynomial (12x² + 74x + 44).

This shows that the set of polynomials is closed under multiplication.

Which is one of the defining properties of a mathematical system.

Therefore, the expression for area of a rectangle is (12x² + 74x + 44) ,

Degree and classification is 2 and quadratic and closure property is demonstrate by product of a polynomial.

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If you can buy 1 bunch of seedless green grapes for $2, then $18 we can buy 9 bunches.

The concept used here is a simple division to find out how many bunches of seedless green grapes can be purchased with a given amount of money.

In this case, we know the cost of one bunch of seedless green-grapes ($2) and we are given a total amount of money ($18) that we can spend on grapes. We can use division to find out how many bunches we can buy with that amount.

⇒ Number of bunches = ($18)/($2) per bunch,

⇒ Number of bunches = 9 bunches;

Therefore, with $18, you can buy 9 bunches of seedless green grapes.

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The coordinate of the resulting point C' after 270-degree rotation clockwise is (-2, -3).

Given that:

Point, C(-3, 2)

Rotation does not change the shape and size of the geometry. But changes the orientation of the geometry.

The coordinate of the resulting point C' after 270-degree rotation clockwise is given as,

C' = (-y, x)

C' = (-2, -3)

The coordinate of the resulting point C' after 270-degree rotation clockwise is (-2, -3).

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The solution of the equation x - √x + 4 - 2√(x+3) =0 is given by x ≈ 0.449 or x = 2.

The expression is equal to,

x - √x + 4 - 2√(x+3) =0

By rearranging the terms and simplifying the expression as follows,

x - √x + 4 - 2√(x+3) = 0

⇒x + 4 = √x + 2√(x+3)

Square both sides to eliminate the square root,

⇒ (x + 4)² = (√x + 2√(x+3))²

⇒ x² + 8x + 16 = x + 4(x+3) + 4√x√x+3

⇒x² + 8x + 16 -x -4x -12 = 4√x√x+3

⇒ x² + 3x + 4 = 4√x√x+3

⇒x² + 3x + 4 = 4√x√(x+3)

⇒ x² + 3x + 4 = 4√(x²+3x)

Squaring both sides again gives,

⇒ (x² + 3x + 4)² = 16(x²+3x)

Expanding the left side gives,

x⁴ + 6x³ + 17x² + 24x + 16 = 16x² + 48x

Simplifying and rearranging the terms,

x⁴ + 6x³ + x² - 24x + 16 = 0

Factor this equation as follows,

(x² + 4x - 2)(x² + 2x - 8) = 0

The solutions are,

x²+ 4x - 2 = 0 or x² + 2x - 8 = 0

Solve each quadratic equation using the quadratic formula,

For x² + 4x - 2 = 0

x = (-4 ± √(4² - 4(1)(-2))) / (2(1))

⇒ x = (-4 ± √24) / 2

Simplifying the expression under the square root gives,

⇒ x = (-4 ± 2√6) / 2

⇒ x = -2 ± √6

For x² + 2x - 8 = 0

x = (-2 ± √(2² - 4(1)(-8))) / (2(1))

⇒x = (-2 ± √36) / 2

Simplifying the expression under the square root gives,

⇒ x = (-2 ± 6) / 2

⇒ x = -4 or x = 2

Discard the negative solution -2-√6 and -4

As it would lead to a negative value under the square root.

The final solutions are,

x ≈ 0.449 or x = 2

Therefore, the solution of the given equation is equal to x ≈ 0.449 or x = 2.

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The above question is incomplete , the complete question is:

Solve for x.

x - √x + 4 - 2√(x+3) =0

Answer:

(5,5π/4)

(5,−3π/4)

(−5,π/4)

(−5,−7π/4)

Step-by-step explanation:The correct answer is:

r = 5 units (distance from the pole)

θ = 5π/4 radians (angle measured counterclockwise from the ray θ = 0)

Note: The polar coordinates for point A on the given graph are r = 5 and θ = 5π/4. The r-value represents the distance from the pole to the point, and the θ-value represents the angle measured counterclockwise from the reference ray θ = 0.

To produce 60 liters of a 55% acid solution, we need 18 liters of a 50% acid solution and 42 liters of a 60% acid solution.

Let x be the number of liters of the 50% acid solution needed, and y be the number of liters of the 60% acid solution needed.

We can set up two equations based on the amount of acid and the total volume:

0.50x + 0.60y = 0.55(60)

x + y = 60

Simplifying the first equation:

0.50x + 0.60y = 33

5x + 6y = 330 (multiplying both sides by 10)

We can solve for one variable in terms of the other in the second equation, and substitute into the first equation:

y = 60 - x

5x + 6(60 - x) = 330

Simplifying:

x = 18

So we need 18 liters of the 50% acid solution, and:

y = 60 - x = 42

So we need 42 liters of the 60% acid solution.

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Answer: (10 * 6) + (9 * 6) = 114

Step-by-step explanation:

Using the distributive property, we can write:

19 × 6 = (10 + 9) × 6

= (10 × 6) + (9 × 6) // Distributive property of multiplication over addition

= 60 + 54

= 114

Therefore, 19 × 6 = 114 when using the distributive law.

The larger cube is made up of 384 small cubes, each with a volume of 1/4.

We have,

If we assume that 96 is the total volume of a larger cube made up of smaller cubes, and each small cube has a volume of 1/4.

We can make the following calculation:

Let x be the number of small cubes that make up the larger cube.

The volume of the larger cube = Volume of each small cube x Number of small cubes

96 = (1/4) x

Solving for x.

Multiply 4 on both sides.

96 x 4 = x

x = 384

Therefore,

The larger cube is made up of 384 small cubes, each with a volume of 1/4.

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Answer:(4,5π/4)

Step-by-step explanation:

Note that the point is on the circle whose radius is labeled 4, so r=4. The angle of the point is 5π4, so θ=5π4. Thus, the polar coordinates are (4,5π4).

The entrepreneur's revenue is  $2325x, entrepreneur's cost is $33,750 + $1700x and entrepreneur's profit from x sold-out performances is  $625x - $33,750

The entrepreneur's revenue from x sold-out performances is given by the formula:

Revenue = (Price per Performance) x (Number of Performances)

Revenue = $2325 x x

Revenue = $2325x

The entrepreneur's cost from x sold-out performances is given by the formula:

Cost = Overhead + (Production Cost per Performance) x (Number of Performances)

Cost = $33,750 + $1700x

The entrepreneur's profit from x sold-out performances is the revenue minus the cost:

Profit = Revenue - Cost

Profit = $2325x - ($33,750 + $1700x)

Profit = $625x - $33,750

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The composition of functions is solved and gof ( -7 ) = 1,286

Given data ,

Let the first function be represented as f ( x )

Now , the value of f ( x ) is

f ( x ) = 13 - 2x

Let the second function be represented as g ( x )

Now , the value of g ( x ) is

g ( x ) = 2x² - 6x - 10

Now , g o f ( x ) = 2 ( 13 - 2x )² - 6 ( 13 - 2x ) - 10

On simplifying , we get

g o f ( x ) = 2 ( 169 + 4x² - 52x ) - 78 + 12x - 10

g o f ( x ) = 250 + 8x² - 92x

Now , the composition of function is

g o f ( -7 ) = 8 ( -7 )² - 92 ( -7 ) + 250

g o f ( -7 ) = 392 + 894

g o f ( -7 ) = 1,286

Hence , the function is g o f ( -7 ) = 1,286

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Without more information or a diagram, it's not possible to definitively determine which side-angle relationship (ASA, AAA, SSS, SAS) the triangle fits. These acronyms represent different combinations of sides and angles known in a triangle.

Given the question, it's clear that we are discussing a concept in Geometry - the side-angle relationships in a triangle. However, without more information or an accompanying diagram, it isn't possible to definitively determine which of the side-angle relationships (ASA, AAA, SSS, SAS) the triangle fits. These acronyms represent different combinations of side and angle information known about a triangle. For example, ASA means that we know two Angles and the Side between them. SAS relates to knowledge of two Sides and the Angle between them, and so forth. More information or a visual representation would allow a more conclusive answer.

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