Estimate and classify the critical points for the graph of the function
Please show work

ax² + bx + c = 0

discriminant = b² - 4ac

----> x² + 7x + 12

---> Discriminant of this equation = 7² - 12.1.4 = 49 - 48 = 1 (Positive discriminant)

x = (-7 + 1)/(2.1) = -6/2 = -3

or

x = (-7 - 1)/(2.1) = -8/2 = -4

Therefore, this equation has two real solutions and has a positive discriminant.

The correct answer is B

Answer:

Step-by-step explanation: 23000 milligrams

Answer:

[tex]A=\frac{\sqrt{3}}{4}a^2[/tex]

Step-by-step explanation:

Since an equilateral has all of the sides equal, we can find the height of triangle using: [tex]a^2+b^2=c^2[/tex]. I attached a diagram which should explain how I got the dimensions of the three sides. Using the information from the diagram we get the equation:

[tex]h^2+(\frac{a}{2})^2=a^2[/tex]

Subtract a^2 from both sides

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

Take the square root of both sides

[tex]h = \sqrt{a^2-(\frac{a}{2})^2}[/tex]

If you know the area of a triangle, it's: [tex]\frac{1}{2}bh[/tex]. In this case the base=a, and the height is what we defined above. Using this we get:

[tex]A = \frac{a}{2}*\sqrt{a^2-(\frac{a}{2})^2}[/tex]

We can distribute the exponent over the division to get:
[tex]A = \frac{a}{2}*\sqrt{a^2-(\frac{a^2}{4})[/tex]

Now we can rewrite a^2 as 4a^2/4

[tex]A = \frac{a}{2}*\sqrt{\frac{4a^2}{4}-(\frac{a^2}{4})[/tex]

Now add the two fractions:

[tex]A = \frac{a}{2}*\sqrt{\frac{3a^2}{4}[/tex]

We can distribute the square root the division just like how we distributed the exponent 2, since the square root can be expressed as an exponent (1/2)

[tex]A = \frac{a}{2}*\frac{\sqrt{3a^2}}{\sqrt{4}}[/tex]

There's a radical identity that states: [tex]\sqrt[n]{a} * \sqrt[n]{b} = \sqrt[n]{a*b}[/tex]. We can use this to rewrite one radical as multiple radicals to simplify it:

[tex]A = \frac{a}{2}*\frac{\sqrt{a^2}*\sqrt{3}}{2}[/tex]

Simplify:

[tex]A = \frac{a}{2}*\frac{a*\sqrt{3}}{2}[/tex]

Now multiply the two fractions

[tex]A = \frac{a^2*\sqrt{3}}{4}[/tex]

This is the formula for the area of an equilateral triangle, but it is also often written as:
[tex]A=\frac{\sqrt{3}}{4}a^2[/tex]

the expression simplified is:

[tex]12x^3 - 25x^2 + 9[/tex]

Here we have the expression:

[tex](4x - 3)*(3x^2 - 4x - 3)[/tex]

We just need to distribute the product, we will get:

[tex](4x - 3)*(3x^2 - 4x - 3)\\\\(4x)*(3x^2 - 4x - 3) - 3*(3x^2 - 4x - 3)\\\\12x^3 - 16x^2 - 12x - 9x^2 + 12x + 9[/tex]

Now we just need to group terms with the same exponent of x:

[tex]12x^3 - 16x^2 - 12x - 9x^2 + 12x + 9\\\\12x^3 + (-16 - 9)x^2 + (-12 + 12)x + 9\\\\12x^3 - 25x^2 + 9[/tex]

That is the expression simplified

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The net number of yards lost by the cowboys football team is; 8 yards.

It follows from the task content that the number of yards gained or lost by the football team during each of the three plays is; gained 7 yards, lost 8 yards and then lost 7 more yard.

Hence, by polarising yard gain as positive and yard loss as negative; we have;

Net total = 7 - 8 -7 = -8

Hence, the team lost 8 yards in the total game.

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Answer:

12

Step-by-step explanation:

Usually, above sea level is represented by a positive number and below sea level by a negative number.

12 ft above sea level = 12

Using the simple interest formula, it is found that the APR for the loan is of 4.472%.

Simple interest is used when there is a single compounding per time period.

The amount of money after t years in is modeled by:

[tex]A(t) = A(0)(1 + rt)[/tex]

In which:

The parameters for this problem are:

A(t) = 6 x 511.18 = 3067.08, A(0) = 3000, t = 0.5.

We solve the equation for r to find the APR.

[tex]A(t) = A(0)(1 + rt)[/tex]

[tex]3067.08 = 3000(1 + 0.5r)[/tex]

[tex]1 + 0.5r = \frac{3067.08}{3000}[/tex]

1 + 0.5r = 1.02236

r = (1.02236 - 1)/0.5

r = 0.04472.

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Answer:

$4.50

Step-by-step explanation:

We can use the distributive property to solve this problem. 0.05(50+40) is the total sales tax. 0.05 being the percentage and 50 and 40 being the things we need to distribute 0.05 to. This becomes 2.5+2. This is 4.5.

Answer:

$0.45

Step-by-step explanation:

0.05(50 + 40) = 50 x 0.05 + 40 x 0.05 = 0.25 + 0.20 = 0.45

The value of the derivative of functions h'(6) as requested in the task content is; 55.

Since it follows from the task content that the function h(x)=4f(x)+5g(x)+1.

Hence, the derivative of h(x) can be evaluated as;

h'(x)=4f'(x)+5g'(x)

On this note, by substitution, it follows that;

h'(6)=4(5)+5(7)

h'(6) = 55.

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Step-by-step explanation:

1) if m(∠A)∈[0;90°), then

[tex]cos(A)=\sqrt{1-sin^2A} =\frac{3}{5};[/tex]

[tex]sin2A=2sinAcosA=2*\frac{3}{5} *\frac{4}{5}=\frac{24}{25};[/tex]

[tex]cos2A=cos^2A-sin^2A=\frac{9}{25}-\frac{16}{25}=-\frac{7}{25};[/tex]

[tex]tan2A=\frac{sin2A}{cos2A}=-\frac{\frac{24}{25}}{\frac{7}{25}}=-\frac{24}{7}.[/tex]

2) if m∠A∈[90°;180°), then

cos(A)=-0.6;

sin2A=-0.96;

cos2A=-0.28;

tan2A=-24/7.

Answer:

f(n)={n/2     = if n is even=   n={2/2

       {3n+1                                 {3(2)+1   =7

Step-by-step explanation:

if n is even u can put any even no in that place and u can get odd no or even no.(i think)

The domain of the rational function is:

{x ∈ ℝ| x ≠ –2, 2}

Here we have the rational function:

[tex]f(x) = \frac{x^2 - x - 12}{x^3 - 4x^2 - 4x + 16}[/tex]

We want to get the domain of that function. First, we can rewrite the numerator and denominator as:

[tex]x^2 -x - 12 = (x + 3)*(x - 4)[/tex]

[tex]x^3 - 4x^2 - 4x + 16 = (x - 4)*(x + 2)*(x - 2)[/tex]

Then we can rewrite the rational function as:

[tex]f(x) = \frac{(x + 3)*(x - 4)}{(x - 4)*(x + 2)*(x - 2)}[/tex]

This can be simplified to:

[tex]f(x) = \frac{(x + 3)}{(x + 2)*(x - 2)}[/tex]

Now, the domain will be the set of all real numbers, minus the values of x that generate problems.

In this case, the values:

x = -2 and x = 2 make the denominator to be zero, and we can't divide by zero, so  we conclue that the domain is:

{x ∈ ℝ| x ≠ –2, 2}

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Answer:

it matches with equation c

Answer:

It is an open circle and you would shade to the right on a number line.

Step-by-step explanation:

The circle is open because you are not including 3.  Put numbers in for t and you will see that the numbers greater than 3 are all on the right side of the number line.

The largest y-value from the solution set of the nonlinear system is 3.78. (Correct choice: C)

In this question we have a system of non-linear equation formed by a radical equation and a quadratic equation, which can be solved both analytically and graphically. By equalizing the two formulas we have the following equation:

√(2 · x + 4) = x² - 5 · x + 3

√(2 · x + 4) + 13 / 4  = x² - 2 · (5 / 2) · x + 25 / 4

√(2 · x + 4) + 13 / 4 = (x - 5 / 2) ²

√(2 · x + 4) = (x - 5 / 2) ² - 13 / 4

Then, we square both sides to eliminate the radical sign:

2 · x + 4 = (x - 5 / 2)⁴ - (13 / 2) · (x - 5 / 2)² + 169 / 16

2 · x + 4 = x⁴ + 4 · x³ · (- 5 / 2) + 6 · x² · (- 5 / 2)² + 4 · x · (- 5 / 2)³ + (- 5 / 2)⁴ - (13 / 2) · [x² + 2 · (- 5 / 2) · x + ( - 5 / 2)²] + 169 / 16

2 · x + 4 = x⁴ - 10 · x³ + (75 / 2) · x² - (125 / 2) · x + 625 /16 - (13 / 2) · x² + (65 / 2) · x - 325 / 8 + 169 / 16

2 · x + 4 = x⁴ - 10 · x³ + 31 · x² - 30 · x + 9

x⁴ - 10 · x³ + 31 · x² - 32 · x + 5 = 0

The real roots of this polynomial are:

x₁ ≈ 5.152, x₂ ≈ 2.862, x₃ ≈ 1.797, x₄ ≈ 0.189

Now we evaluate f(x) at each root:

y₁ ≈ 3.782, y₂ ≈ 3.118, y₃ ≈ 2.756, y₄ ≈ 2.092

The largest y-value from the solution set of the nonlinear system is 3.78. (Correct choice: C)

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This snow flake-like figure can be generated by rotating an end 60° five times around the center of the hexagon. There are two forms: (i) clockwise, (ii) counterclockwise.

Geometrically speaking, snow flakes represent regular hexagons. Regular hexagons can divided into six concentric regular triangles, whose central angles have a measure of 60°. This fractal figure can be generated by rotating 60° five times around the center.

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Answer: 51.4°

Step-by-step explanation:

Use law of cosines:

(because you are trying to find angle B, you need to use Cos B)

Cos B = c² + a²-b²/ 2ca

Then, plug in your numbers:

 Cos B = 30²+64²-51²/ 2(30)(64)

Simplify:

Cos B = 0.6237

Next, to get rid of Cos B, so that we have just B, you need to do the Arccosine or 0.6237:

B = arccosine (0.6237)

Which gets you: 51.4°    

The measure of angle B is 51.41 degrees after applying the cos law of the triangle.

In terms of geometry, the triangle is a three-sided polygon with three edges and three vertices. The triangle's interior angles add up to 180°.

We have given a triangle in the figure:

To find the measure of the angle B

We can apply cos law:

c² = a² + b² - 2ab cosB

The a, b, and c represents the side lengths of the triangle and the measures are:

a = 30

b = 64

c = 51

51² = 30² + 64² - 2(30)(64) cosB

3840CosB = 4996 - 2601

3840CosB = 2395

CosB = 2395/3840

CosB = 0.623

B = Cos⁻¹(0.623)

B = 51.41 degrees

Thus, the measure of angle B is 51.41 degrees after applying the cos law of the triangle.

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Answer: -5=x

Step-by-step explanation:

kmbtvonpvnp4tnv

Answer:

we need a whole example not just the variables.

Step-by-step explanation:

The algebraic description that maps the image ΔABC onto ΔA′B′C′ is (x, y) ⇒ (x + 7, y - 4)

Transformation is the movement of a point from its initial location to a new location. Types of transformations are reflection, rotation, translation and dilation.

Translation is the movement of a point either up, left, right or down in the coordinate plane.

The algebraic description that maps the image ΔABC onto ΔA′B′C′ is (x, y) ⇒ (x + 7, y - 4)

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The price that will maximize company’s revenue of $10,000 is $50.

From the information, the company determines 50that the revenue, in dollars, for selling a particular model of lamp is given by R(x)=x(−20x+1200),

Therefore, this will be:

R(x) = -20x² + 1200x

10000 = -20x² + 1200x

-20x² + 1200x - 10000 = 0

x² - 60x + 500

x² - 50x - 10x + 500

x(x - 50) - 10(x - 50)

(x - 50)(x - 10)

Therefore, x - 50 = 0

x = 0 + 50 = 50

The price is 50.

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90 + 6x = 180

-90

6x = 90

÷6

x = 15

5y + 4y + 90 + (6x15) =360

9y + 90 + 90 = 360

9y + 180 = 360

- 180

9y = 180

÷9

y = 20

Hope this helps!

The ratio between 28,000 and 14 is 2000 : 1

The numbers are given as:

28,000 and 14

Express as ratio

Ratio = 28000 : 14

Divide each number by 14

Ratio = 2000 : 1

Hence, the ratio between 28,000 and 14 is 2000 : 1

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The slope of the line passing through the points (-4, -9) and (5, -9) is 0

The points are (-4, -9) and (5, -9).

The slope is calculated using:

Slope = (y2 - y1)/(x2 - x1)

So, we have:

Slope = (-9 + 9)/(5 + 4)

Evaluate

Slope = 0

Hence, the slope of the line passing through the points (-4, -9) and (5, -9) is 0

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The equations which represents the minimum and maximum scores of the students is; x = 87 ± 1.8.

According to the task content, the

Average score of the students in the science class is; 87 points

All students are within 1.8 points of the average,

it therefore follows from the task content that the

minimum score is;

= 87 -1.8

= 85.2

maximum score is;

= 87+1.8

= 88.8 points.

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Answer:

Step-by-step explanation:

A. |x − 87| = 1.8

-hope this helps.

Answer:

58

Step-by-step explanation:

[tex]x = \frac{141 - 25}{2} = 58[/tex]

The cube can contain 1000 cm^3 of water

The side length is given as:

L = 10 cm

The amount of water it can contain is calculated as;

V =L^3

So, we have:

V = 10^3

Evaluate

V = 1000

Hence, the cube can contain 1000 cm^3 of water

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The equation of the function g(x) is g(x) = 7^(3x) + 1

The function f(x) is given as:

f(x) = 7^x  1

When compressed horizontally by a factor of 1/3.

We have:

g(x) = f(3x)

This implies that:

g(x) = 7^(3x) + 1

Hence, the equation of the function g(x) is g(x) = 7^(3x) + 1

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Answer: 36

Step-by-step explanation:

As this is a two-digit number, it will have a digit in its 10's place and a digit in its 1's place. We can represent the 10's digit using x and the 1's digit using y.

Since the ones is twice the number of tens, this means that y is equal to 2*x. Let's put this into an equation.

[tex]y=2x[/tex]

If the 10's digit is x and the 1's digit is y, the original number would be [tex]10x + y[/tex], as x must be multiplied by 10 to get it to the 10's place. Similarly, the reversed number would be [tex]10y + x[/tex], as y must be multiplied by 10 to get it to the 10's place.

Twice the original number (10x+y) is equal to 9 more than (i.e. 9 plus) the reversed number. We can put this into an equation to help us answer the question.

[tex]2(10x+y)=9+10y+x\\20x+2y=9+10y+x\\19x+2y=9+10y\\19x=9+8y[/tex]

Now we have a system of two equations.

[tex]y=2x\\19x=9+8y[/tex]

We can solve this system by substituting y for 2x in the second equation. Then, we can isolate and get the value of x.

[tex]19x=9+8(2x)\\19x=9+16x\\3x=9\\x=3[/tex]

Now that we have the value of x, let's put it back into the first equation and solve for y.

[tex]y=2(3)\\y=6[/tex]

Remember that x is the 10's digit and y is the one's digit of our answer. Since x is 3 and y is 6, our answer is 36.

We can quickly check if our answer is right by making sure both conditions in the question are met.

6 (the one's digit) is twice the value of 3 (the ten's digit), making the first condition true. Twice of 36 is 72, which is 9 more than the reversed number, which is 63.