Find the greatest common monomial factor : 4a^5b^2 and ab

The solution to the system of equations are ±2i and ±i

Given the quartic equation

x^4 + 3x^2 + 2 = 0

Let u = x^2 to have:

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

u^2 + 3u + 2 = 0

Factorize

u^2 + u +2u + 2 = 0

u(u + 1) + 2(u + 1) = 0

(u + 2)(u + 1) = 0

u = -2 or -1

If u = x^2

-2 = x^2

x = √-2

x = 2i

Similarly

If u = x^2

-1 = x^2

x = √-1

x = i

Hence the solution to the system of equations are ±2i and ±i

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Final answer:

To solve the equation x^4 + 3x^2 + 2 = 0 using u substitution, substitute x^2 for u and solve the quadratic equation. The resulting solutions in the complex number system are x = i, -i, √2i, and -√2i. There are no real solutions for this equation.

Explanation:

The solutions of the equation x4 + 3x2 + 2 = 0 can be found using u substitution. First, let's substitute u for x2, which transforms the original equation into the form u2 + 3u + 2 = 0. This is a quadratic equation, and we can solve it using the quadratic formula or by factoring. Factoring (u + 1)(u + 2) = 0, we get the solutions for u as u = -1 and u = -2. Since u is a substitute for x2, we then solve for x.

For u = -1, the equation x2 = -1 has no real solutions because the square of a real number cannot be negative. However, in the complex number system, the solutions are x = i and x = -i, where i is the square root of -1.

For u = -2, the equation x2 = -2 also has no real solutions, but the complex solutions are x = √2i and x = -√2i.

Therefore, the full set of solutions for the original equation are x = i, -i, √2i, and -√2i.

To solve for 'x' in the parallelogram area equation, we multiplied the given base (5 units) by the height (x + 6 units) and set it equal to the given area (20 square units). After simplifying, we found that x must be -2 to satisfy the equation 5x + 30 = 20.

The question presented is a typical algebra problem where we need to find the value of 'x' in the context of the area of a parallelogram. The base of the parallelogram is given as 5 units and its height is given as (x + 6) units, with the total area given as 20 square units. Since the formula for the area of a parallelogram is base × height, we can set up the equation 5 × (x + 6) = 20 to find the value of x.

Now, we will solve for x:

Hence, the correct answer is b. x=-2.

Answer:

B. 4

Step-by-step explanation:

This table that shows the input and output values for an exponential function f(x) is,

x                -3          -2       -1      0      1      2       3

f(x)          1/256      1/64    1/16   1/4    1     4       16

Taking [tex]x=2[/tex] and [tex]x=3[/tex] as input (because they are 1 apart), the ratio of output is,

[tex]=\dfrac{f(3)}{f(2)}[/tex]

[tex]=\dfrac{16}{4}[/tex]

[tex]=4[/tex]

Answer:

The correct answer is B. 4

Step-by-step explanation:

Answer:

{y|−5≤y≤1}

Step-by-step explanation:

The range of the function is y ∈ [-5, 1] or  -5 ≤ y ≤ 1 if the function is  y = 3cos4x - 2 option second is correct.

It is defined as a special type of relationship, and they have a predefined domain and range according to the function every value in the domain is related to exactly one value in the range.

We have a function:

y = 3cos4x - 2

As we know, cosx ∈ [-1, 1]

-1 ≤ cosx ≤ 1

-1 ≤ cos4x ≤ 1

-3 ≤ 3cos4x ≤ 3

-5 ≤ 3cos4x - 2 ≤ 1

-5 ≤ y ≤ 1

Thus, the range of the function is y ∈ [-5, 1] or  -5 ≤ y ≤ 1 if the function is  y = 3cos4x - 2 option second is correct.

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Answer:

1

Step-by-step explanation:

Answer is 1 for Apex Precal

The movie theater sold 117 adult tickets and 46 child tickets.

To solve this problem, we can set up a system of equations. Let's say the number of adult tickets sold is x and the number of child tickets sold is y. We can then write two equations based on the given information:

We can solve this system of equations by either substitution or elimination. Let's use elimination:

6x + 6y = 978

(6x + 6y) - (11x + 6y) = 978 - 163

-5x = -585

Divide by -5:

x = 117

117 + y = 163

y = 163 - 117

y = 46

Therefore, 117 adult tickets and 46 child tickets were sold.

The [tex]x[/tex] coordinates of the point A is [tex]\boxed{\bf 0}[/tex].

The [tex]x[/tex] coordinates of the point B is [tex]\boxed{\bf 0}[/tex].

The [tex]x[/tex] coordinates of the point C is [tex]\boxed{\bf 10}[/tex].

Further explanation:

Given:

The coin labeled as A lies on the [tex]y[/tex]-axis.

The coin labeled as B lies on the origin.

The coin labeled as C lies on the [tex]x[/tex]-axis.

The distance of the point B and C is [tex]10\text{ cm}[/tex].

Concept used:

The point which lies on the [tex]x[/tex]-axis, the value of its [tex]y[/tex]-coordinate is [tex]0[/tex].

The point which lies on the [tex]y[/tex]-axis, their [tex]x[/tex]-coordinate is [tex]0[/tex].

The coordinate of the origin is [tex](0,0)[/tex].

Calculation:

The coin labeled as A lies on the [tex]y[/tex]-axis therefore the [tex]x[/tex]-coordinate of the point is [tex]0[/tex].

The coin labeled as B lies on the origin therefore the [tex]x[/tex]-coordinate of the point is [tex]0[/tex].

The distance from point B to the point C is [tex]10\text{ cm}[/tex] and the coin labeled as C lies on the x axis it means that the [tex]x[/tex]-coordinate of the point is [tex]10[/tex].  

Therefore, the [tex]x[/tex] coordinate of the point A is [tex]0[/tex].

The [tex]x[/tex] coordinate of the point B is [tex]0[/tex].

The [tex]x[/tex] coordinate of the point C is [tex]10[/tex].

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Answer details:

Grade: High school

Subject: Mathematics

Chapter: Coordinate geometry

Keywords: Coordinate geometry, x-axis, y-axis, x-coordinate, y-coordinate, coin, square, three corners, three identical coins.

The distance from Honolulu to Turtle Bay is approximately 37 miles. To calculate this distance, you can use a map or a GPS tool. The most direct route between the two locations is along the Kamehameha Highway.

The distance from Honolulu to Turtle Bay is approximately 37 miles. To calculate this distance, you can use a map or a GPS tool. The most direct route between the two locations is along the Kamehameha Highway.

It's important to note that the actual distance traveled may vary depending on the specific route taken and any detours or alternate paths chosen.

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Answer:

a) (3x+4)(9x^2-12x+16)

Step-by-step explanation:

Answer: Option 'B' is correct.

Step-by-step explanation:

Since we have given that

Dimensions of the photo is given by

[tex]20\ in.\ by\ 24\ in.[/tex]

According to question, the photo is reduced so that the longer dimension is 15 inch.

So, Let the width of the reduced photo be x.

So, it becomes,

[tex]\frac{20}{x}=\frac{24}{15}\\\\x=\frac{15\times 20}{24}\\\\x=12.5\ in.[/tex]

So, the width of the reduced photo is 12.5 inch.

Hence, Option 'B' is correct.

Answer/Step-by-step explanation:

149.99+(149.99*0.07)=160.4893. Round to $160.49

Final answer:

The values of k that ensure the points (-5, 5) and (k, 0) are [tex]\(\sqrt{29}\)[/tex]units apart are -3 and -7, derived using the Pythagorean theorem and algebraic manipulation.

Explanation:

To find all values of k so that the points (-5, 5) and (k, 0) are \(\sqrt{29}\) units apart, we use the distance formula derived from Pythagoras’ theorem, which is[tex]d = \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)[/tex]. Here, [tex]\(x_1 = -5\), \(y_1 = 5\), \(x_2 = k\), and \(y_2 = 0\). Substituting the given values, we get:[/tex]

[tex]\(\sqrt{29} = \sqrt{(k + 5)^2 + (0 - 5)^2}\)[/tex]

Squaring both sides to eliminate the square root gives:

[tex]29 = (k + 5)^2 + 25[/tex]

Subtracting 25 from both sides, we get:

[tex]4 = (k + 5)^2[/tex]

Therefore, \(k + 5 = \pm2\), which simplifies to:

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

[tex]\(k + 5 = -2\) giving \(k = -7\)[/tex]

The values of k are -3 and -7, so the correct option is c. -3, -7.

The correct answer is option (c) -3, -7.

How is it so?

To find the values of k such that the given points (-5, 5) and (k, 0) are [tex]\(\sqrt{29}\)[/tex] units apart, use the distance formula between two points in a plane:

[tex]\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]

Given that the distance between the points is [tex]\(\sqrt{29}\)[/tex], we have:

[tex]\[ \sqrt{(k - (-5))^2 + (0 - 5)^2} = \sqrt{29} \][/tex]

[tex]\[ \sqrt{(k + 5)^2 + 5^2} = \sqrt{29}[/tex]]

[tex]\[ \sqrt{k^2 + 10k + 25 + 25} = \sqrt{29} \][/tex]

[tex]\[ \sqrt{k^2 + 10k + 50} = \sqrt{29} \][/tex]

Squaring both sides to eliminate the square root:

[tex]\[ k^2 + 10k + 50 = 29 \][/tex]

[tex]\[ k^2 + 10k + 21 = 0 \][/tex]

Solve this quadratic equation

[tex]\[ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

Where a = 1, b = 10, and c = 21.

[tex]\[ k = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 1 \cdot 21}}{2 \cdot 1} \][/tex]

[tex]\[ k = \frac{-10 \pm \sqrt{100 - 84}}{2} \][/tex]

[tex]\[ k = \frac{-10 \pm \sqrt{16}}{2} \][/tex]

[tex]\[ k = \frac{-10 \pm 4}{2} \][/tex]

[tex]\[ k = \frac{-10 + 4}{2} \] or \( k = \frac{-10 - 4}{2} \)[/tex]

[tex]\[ k = \frac{-6}{2} \] or \( k = \frac{-14}{2} \)[/tex]

[tex]\[ k = -3 \] \:or \( k = -7 \)[/tex]

So, the correct answer is option (c) -3, -7.

circumference of a circle= pi multiplied by the diameter

= 20ft x pi

=20 pi feet so the answer is b

Answer:

[tex]2\sqrt{5}[/tex]

C is the correct option.

Step-by-step explanation:

The given expression is [tex]\sqrt{180}-\sqrt{125}+\sqrt{5}[/tex]

In order to simplify the expression, we find the factors.

180 =6 ×6×5

125 = 5 ×5×5

Hence, we can rewrite the expression as

[tex]\sqrt{6\times6\times5}-\sqrt{5\times5\times5}+\sqrt{5}[/tex]

We can further write this expression as

[tex]\sqrt{6^2\times5}-\sqrt{5^2\times5}+\sqrt{5}[/tex]

Now, we can use the rule [tex]\sqrt{x^2}=x[/tex]

[tex]6\sqrt{5}-5\sqrt{5}+\sqrt{5}[/tex]

Finally, we can combine these like terms

[tex]2\sqrt{5}[/tex]

Hence, C is the correct option.