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vfiekz
1 month ago
13

The most important component of a physical structure is a solid foundation on which the rest of the structure sits. A strong fou

ndation allows buildings to last longer and withstand many outside forces.
Just like the foundation of a building, a strong foundation for algebra is critical for success and allows you to extend your understanding to more complex mathematical and real-world situations. What are some of the foundational, or key, concepts of algebra that you have learned so far?
Mathematics
1 answer:
tester [12.3K]1 month ago
6 0

Answer and explanation:

Algebra revolves around the fundamental idea of using letters known as variables to represent quantities, which allows for solving for unknown values. Essentially, algebra involves transitioning from what is known to what is unknown to ascertain those unknown results. For instance, if we know a specific item was purchased twice but we're unsure of its price, we can denote this unknown price as 2a or 2p, depending on the selected variable. If the total spending for those items is, say, $50, we can set up the equation 2a = $50, which leads us to find that the cost per item is $25.

Algebra can also manifest itself in expressions, commonly referred to as algebraic expressions, which can be incorporated into equations, such as the previously mentioned 2a = $50. These expressions may take forms like 2a + 3b, where a and b designate the costs of different products that were acquired in quantities of 2 and 3, respectively.

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An advertising company designs a campaign to introduce a new product to a metropolitan area of population 3 Million people. Let
PIT_PIT [12445]
Step-by-step explanation: Since the number P(t) increases proportionally to the number of individuals unaware of the product, we establish that initially, nobody is aware of the product at the campaign's start and that 50% are aware after 50 days of advertisements. Thus, we determine: P(0) = 0 and P(50) = 1,500,000, leading to a first-order ordinary differential equation. The integrating factor must be calculated and both sides of the equation manipulated accordingly. Hence, upon integrating and solving, we arrive at the equation modeling the number of people (in millions) aware of the product over time.
5 0
1 month ago
Two functions are shown in the table below. Function 1 2 3 4 5 6 f(x) = −x2 + 4x + 12 g(x) = −x + 6 Complete the table on your o
Svet_ta [12734]

For \fbox{\begin \\\math{x}=6\\\end{minispace}} the function f(x)=-x^{2} +4x+12 and g(x)=-x+6 both yield the same result.

Detailed breakdown:  

The functions involved are

f(x)=-x^{2}+4x+12

g(x)=-x+6

Step 1:  

Insert x=1 in f(x)=-x^{2} +4x+12 to find the value of f(1).

f(1)=-1^{2} +4(1)+12\\f(1)=-1+4+12\\f(1)=15

Insert x=1 in g(x)=-x+6 to find the value of g(1).

g(1)=-1+6\\g(1)=5

Step 2:

Insert x=2 in f(x)=-x^{2} +4x+12 to obtain the value of f(2).

f(2)=-2^{2} +4(2)+12\\f(2)=-4+8+12\\f(2)=16

Substitute x=2 into g(x)=-x+6 to find the value of g(2).

g(2)=-2+6\\g(2)=4

Step 3:

Replace x=3 in f(x)=-x^{2} +4x+12 to find the value of f(3).

f(3)=-3^{2} +4(3)+12\\f(3)=-9+12+12\\f(3)=15

Also, replace x=3 in g(x)=-x+6 to find the value of g(3).

g(3)=-3+6\\g(3)=3

Step 4:

Insert x=4 in f(x)=-x^{2} +4x+12 to find the value of f(4).

f(4)=-4^{2} +4(4)+12\\f(4)=-16+16+12\\f(4)=12

Also, replace x=4 in g(x)=-x+6 to obtain the value of g(4).

g(4)=-4+6\\g(4)=2

Step 5:

Insert x=5 in f(x)=-x^{2} +4x+12 to obtain the value of f(5).

f(5)=-5^{2} +4(5)+12\\f(5)=-25+20+12\\f(5)=7

Replace x=5 in g(x)=-x+6 to find the value of g(5).

g(5)=-5+6\\g(5)=1

Step 6:

Insert x=6 into f(x)=-x^{2} +4x+12 to find the value of f(6).

f(6)=-6^{2} +4(6)+12\\f(6)=-36+24+12\\f(6)=0

Also, substitute x=6 in g(x)=-x+6 to obtain the value of g(6).

g(6)=-6+6\\g(6)=0

Step 7:

According to the provided condition f(x)=g(x).

(a). Insert f(x)=-x^{2} +4x+12 and g(x)=-x+6 into the previously mentioned equation.

-x^{2} +4x+12=-x+6

(b). Multiply through by -1 on both sides.

x^{2} -4x-12=x-6

(c). Move the term x-6 to the left side of the equation.

x^{2} -4x-12-x+6=0\\x^{2} -5x-6=0

(d). Divide the middle term so that its sum equals 5 and the product equals 6.

x^{2} -(6-1)x-6=0\\x^{2} -6x+x-6=0\\x(x-6)+1(x-6)=0\\(x+1)(x-6)=0\\x=-1,6

From the analysis above, it is noted that for x=6 both functions f(x) and g(x) yield the same outcome.

Using a direct approach:

f(x)=g(x)\\\Leftrightarrow-x^{2} +4x+12=-x+6\\\Leftrightarrow-x^{2} +4x+12+x-6=0\\\Leftrightarrow-x^{2} +5x+6=0\\\Leftrightarrow-x^{2} +6x-x+6=0\\\Leftrightarrow x^{2} -6x+x-6=0\\\Leftrightarrow x(x-6)+1(x-6)=0\\\Leftrightarrow(x+1)(x-6)=0\\\Leftrightarrow x=6,-1

The table representing function f(x)=-x^{2} +4x+12 and g(x)=-x+6 is included below.

For more information:

1. What is the y-intercept of the quadratic function f(x) = (x – 6)(x – 2)? (0,–6) (0,12) (–8,0) (2,0)

2. Which is the graph of f(x) = (x – 1)(x + 4)?

6 0
22 days ago
In a class of 30 students, 19 are studying French (F), 12 are studying Spanish (S) and 7 are studying both French and Spanish. H
lawyer [12517]

Answer: Fourteen students are not enrolled in any foreign languages.

Step-by-step explanation: Begin with the total student count and subtract the number enrolled in each language class. This yields 19 for French, 12 for Spanish, and 7 for those taking both. Summing these numbers gives 38. Next, repeat this process, subtracting each category from 30 to find how many students are not participating in either language. You would end up with 11 for French, 18 for Spanish, and 23 for both. Adding these values results in 52. Finally, subtract 38 from 52 to arrive at the final result of 14.

5 0
24 days ago
A rancher’s herd of 250 sheep grazes over a 40-acre pasture. He would like to find out how many sheep are grazing on each acre o
AnnZ [12381]

Response:

0.16 acres for each sheep

Explanatory steps:

With a total of 40 acres allocated to 250 sheep, you would calculate the area per sheep by dividing the total acres by the sheep count. This can be represented as:

40/250 =....

7 0
1 month ago
Read 2 more answers
The parent function f(x) = log3x has been transformed by reflecting it over the x-axis, stretching it vertically by a factor of
babunello [11817]
The third option is correct. Step-by-step explanation: Various transformations apply to a function f(x). If a transformation is applied downward by 'k' units, the function shifts down; if upward, it rises 'k' units. Additionally, if scaled vertically by a factor of 'b', it will stretch; if reflected over the x-axis, the operation is indicated. Thus, since the parent function has undergone reflection over the x-axis, a vertical stretch by a factor of 2, and a downward shift of three units, we can derive that the transformed function is presented.
3 0
24 days ago
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