Mark's father travels 463 miles every month for his job. How many miles does he travel in 8 months?

Solution :

To factor completely

[tex] a^{2} + 3a-28 [/tex]

To factor it completely , first take the product of first and third term, and then break the second term in two parts in such a way that its sum equals the second term and the product equals the product of first and third term.

Product of first and third term is [tex] (a^{2})(-28) = -28 a^{2} [/tex]

Second term is 3a, Re-write 3a as sum of 7a and -4a.

Lets check it Product of 7a and -4a is [tex] -28 a^{2} [/tex] wich is eaqual to product of first and third term and sum of 7a and -4a is 3a which is the second term.

Now factorise, we get

[tex] a^{2}+3a-28\\\\ = a^{2} +7a-4a-28\\\\=a(a+7)-4(a+7)\\\\=(a+7)(a-4) [/tex]

Hence, [tex] (a+7)(a-4) [/tex] is the factor of [tex] a^{2}+3a-28 [/tex].

Final answer:

To find the three consecutive odd integers whose sum is 327, we set up an equation with n as the first odd integer, resulting in the integers being 107, 109, and 111.

Explanation:

The question asks for the three consecutive odd integers whose sum is equal to 327. To solve this, we can represent the first integer as n, the second as n+2, and the third as n+4 because odd numbers are two units apart. We can then set up the following equation to find the value of n:

n + (n + 2) + (n + 4) = 327

Simplifying this equation, we get:

3n + 6 = 327

Subtracting 6 from both sides gives us:

3n = 321

Dividing both sides by 3, we find that n = 107. Therefore, the three consecutive odd integers are 107, 109, and 111.

It is given that on Tuesday, apples were on sale for 20% off. Now, we know that on Monday Mrs. Wise bought 3 lbs of apples at $4 per lb. This means that if we have a 20% off on Tuesday then this must be with respect to the price on Monday which is $4. Thus the price of Tuesday can be calculated as:

[tex] 4-\frac{20}{100}\times 4=4-0.8=3.2 [/tex]

Thus, the per pound price of apples on Tuesday was $3.2

Answer:

d

Step-by-step explanation:

The square root of -12, an imaginary number, can be simplified to 2i√3. This involves breaking -12 into -1 * 12 and taking the square root of both parts to get 'i' and 2i√3 respectively.

The question asked is to simplify the imaginary number square root of -12. Imaginary numbers are those numbers whose square is less than or equal to zero. They are expressed in the form of 'i' where 'i' is the imaginary unit with the property i^2 = -1. When we take square root of a negative number, 'i' is involved in the solution.

To simplify the square root of -12, we can express -12 as -1 * 12. In this case, square root of -1 is 'i' and square root of 12 is 2 * square root of 3 (since 12= 4*3 and square root of 4 is 2).

Therefore, square root of -12 can be written as 'i' * 2 * square root of 3 or simply 2i√3.

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Answer:

5ft

Step-by-step explanation:

The required greatest distance, in feet, a person could be from the lamp and be detected is 5 ft. Option A is correct.

The circle is the locus of a point whose distance from a fixed point is constant i.e center (h, k). The equation of the circle is given by

(x - h)² + (y - k)² = r²

where h, k is the coordinate of the center of the circle on the coordinate plane and r is the radius of the circle.

Here,
Given equation

(x+15)²+(y−12)²=25

The above equation is the equation of the circle, so the longest distance that person could be from the lamp and be detected is the radius of the circle,

Which is given as,
r² = 25
r = √25
r = 5

Thus, the required greatest distance, in feet, a person could be from the lamp and be detected is 5 ft. Option A is correct.

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Final answer:

To solve the equation for a, subtract z from both sides, factor out a, and then divide by (m - b), resulting in a = -z / (m - b), assuming m and b are not equal.

Explanation:

To solve the equation z + ma = ba for a, we need to isolate a on one side of the equation. We can rewrite the equation, moving all terms involving a to one side and all other terms to the opposite side. This process is called collecting like terms.

Here are the steps to isolate a:

Subtract z from both sides of the equation to get ma = ba - z.

Since both terms on the right hand side involve a, factor a out: a ( m - b ) = -z.

Finally, divide both sides of the equation by (m - b) to solve for a: a = -z / (m - b), assuming that m ≠ b.

This algebraic manipulation gives us the value of a in terms of z, m, and b.

Answer:

The slope intercept form of the required line is [tex]y=\frac{-3}{2}x[/tex].

Step-by-step explanation:

If a line is defined as

[tex]Ax+By+C=0[/tex]        ... (1)

Then the slope of the line is

[tex]m=\frac{-A}{B}[/tex]

The given equation is

[tex]3x+2y-6=0[/tex]       .... (2)

From (1) and (2), we get

[tex]A=3, B=2, C=-6[/tex]

The slope of the line is

[tex]m=\frac{-3}{2}[/tex]

The slope of parallel line is same. So, the slope of required line is -3/2.

The slope intercept form of a line is

[tex]y=mx+b[/tex]

Where, m is slope and b is y-intercept.

The slope of required line is -3/2 and y-intercept is at (0,0).

[tex]y=\frac{-3}{2}x+0[/tex]

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

Therefore the slope intercept form of the required line is [tex]y=\frac{-3}{2}x[/tex].

Answer:

Trapzoid

Step-by-step explanation:

Given: A (-2, 3), B (9, 3), C (5, 6) and D (2, 6)

We will find the slope of each line.

Formula:

[tex]\text{Slope, m}=\dfrac{y_2-y_1}{x_2-x_1}[/tex]

[tex]\text{Slope of AB, m}_1=\dfrac{3-3}{9+2}=0[/tex]

[tex]\text{Slope of BC, m}_2=\dfrac{6-3}{5-9}=-\dfrac{3}{4}[/tex]

[tex]\text{Slope of CD, m}_3=\dfrac{6-6}{2-5}=0[/tex]

[tex]\text{Slope of AD, m}_4=\dfrac{6-3}{2+2}=\dfrac{3}{4}[/tex]

Slope of AB = Slope of CD = 0

[tex]m_1=m_3=0[/tex]

Thus, AB is parallel to CD

Slope of BC ≠ Slope of AD

[tex]m_2\neq m_4[/tex]

Thus, BC is not parallel to AD

The quadrilateral ABCD has two sides are parallel and two are not parallel.

Hence, The quadrilateral is trapzoid

To prove similarity, the triangles must have three angles or any three components must be equal. According to the statement, option A seems to be proof the triangles are similar to each other.

There are two images given in the question.

Let us segregate the images into image 1 and image 2.

Take image 1.

In image 1, three different triangles are given,

According to the above bullet points, image 1 qualifies all the requirements for the similarity of the triangles.

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The provided expression with 'z = 36' is incomplete and lacks an operation or additional numbers, making it impossible to calculate its value. The student is advised to provide the full expression for proper assistance.

The student asked: What is the value of this expression when z = 36?

Unfortunately, there is a typographical error in the expression provided. It seems to be missing an operation or additional numbers. Given the format '/z', it seems like it should have been a mathematical operation involving the variable 'z'. However, without an operator such as '+' (plus), '-' (minus), '*' (multiply), or '/' (divide), we cannot determine what the expression is supposed to be or calculate its value given z = 36.

If this is an expression from a specific textbook or sheet, I recommend double-checking the source material for the correct expression. If it's a part of a larger problem or context, please provide the full details so that an accurate computation can be made.

Solving the quadratic equation 2x² + 3x = - 8 for x, x = - 3/4 ± i√55/4

To solve 2x² + 3x = - 8, we proceed as follows.

Since we have the quadratic equation 2x² + 3x = - 8, to solve by completing the square, we rewrite the equation as

2x²/2 + 3x/2 = - 8/2

x² + 3x/2 = - 4

Next, we add the square of half the coefficient of x to both sides. So, we have that

x² + 3x/2 + (3/2 ÷ 2)² = - 4 + (3/2 ÷ 2)²

x² + 3x/2 + (3/2 × 1/2)² = - 4 + (3/2 × 1/2)²

x² + 3x/2 + (3/4)² = - 4 + (3/4)²

(x + 3/4)² = - 4 + 9/16

(x + 3/4)² = (-64 + 9)/16

Now, taking the square root of both sides, we have that

(x + 3/4)² = -55/16

√(x + 3/4)² = ±√(-55/16)

x + 3/4 = ±√-55/4

x + 3/4 = ±i√55/4

x = - 3/4 ± i√55/4

So, the value of x = - 3/4 ± i√55/4