Describe how to derive the quadratic formula from a quadratic equation in standard form. edginuity

Final answer:

The statement that the product of (a − b)(a − b) equals [tex]a^2 - b^2[/tex] is never true; the correct product is always [tex]a^2 - 2ab + b^2[/tex], representing the square of a difference formula.

Explanation:

The statement "The product of (a − b)(a − b) is  [tex]a^2 - b^2[/tex]." is never true. Instead, using the distributive property to expand the expression (a − b)(a − b) yields a different result, as each term in the first binomial must be multiplied by each term in the second binomial.

The correct expansion is:

Thus, the product of (a − b)(a − b) is always  [tex]a^2 - 2ab + b^2[/tex], which is the square of a difference formula, not  [tex]a^2 - b^2[/tex] which is the difference of squares formula. The latter is applicable to the product (a + b)(a - b).

It is found that the composition of injective functions is injective.

An injective function is not a surjective function. This means that we want a function that has a unique output for each input, that doesn't cover the natural numbers.

Proof 1: Proving the composition of injective functions is injective.

f(x) is injective

f(a) = f(b)

a = b

Thus g(x) is injective

g(a) = g(b)

a = b

To prove; f(g(x)) is injective

f(g(a)) = f(g(b))

f(a) = f(b)

Since, g(x) is injective

a = b

Therefore, if f(g(a)) = f(g(b)), then a = b

That proving g(f(x)) is injective is going to follow a similar path of steps.

Proof 2: Proving the composition of surjective functions is surjective.

f(x) is surjective

The range of the function of x is the set of all real numbers

g(x) is surjective

The range of the function of x is the set of all real numbers

f(g(x)) is also surjective. The range of f(x) is the set of all real numbers. where g(x) represents all reals as well means that we touch all of the inputs needed to generate all the outputs for f(x).

Proving g(f(x)) to be surjective follows a similar path.

Learn more about function here:

https://brainly.com/question/2253924

#SPJ5

Answer:

The correct option is B.

Step-by-step explanation:

The slope of a line is constant.

If a line passes through two points then the slope of the line is

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

It is given that the line passes through (-2,7) and (a,2).

Using the above formula the slope of the line is

[tex]m=\frac{2-7}{a-(-2)}[/tex]

[tex]m=\frac{-5}{a+2}[/tex]

It is given that the slope of the line is -5. Substitute m=-5 in the above equation.

[tex]-5=\frac{-5}{a+2}[/tex]

On cross multiplication we get

[tex]-5(a+2)=-5[/tex]

Divide both sides by -5.

[tex]a+2=1[/tex]

Subtract 2 from both the sides.

[tex]a=1-2[/tex]

[tex]a=-1[/tex]

The value of a is -1. Therefore the correct option is B.

Answer:

The answer is B.

Step-by-step explanation:

The mechanic's hourly rate is $55.

Understanding Labor Costs in a Repair Shop

To determine the hourly rate a repair shop charges for labor, we must analyze the information given and apply simple arithmetic operations. According to the examples provided, an auto mechanic charges a flat fee and an hourly rate for labor. For instance, a mechanic charging a flat fee of $75 per car, plus an hourly rate of $55 can be represented by the linear equation y = 75 + 55x, where 'y' is the total labor charge, and 'x' is the number of hours worked.

Example Calculation

Using the equation above, if the job takes 3.5 hours, the labor charge (y) would be calculated as $75 + ($55  imes 3.5) = $75 + $192.50 = $267.50. Hence, we can infer that the mechanic's hourly rate is $55.

before the bet Rob had 7 out of 11 ( 4+7)

280/7 = 40

after bet Rob had 3

3*40 = 120

 he has $120

The answer is C. [1, ∞] on edge

To determine how long it will take for the population to triple, we need to solve the equation 3P0 = P0e^(rt) for t. Rounding the answer to one decimal place, it will take approximately 10.1 years for the population to triple.

To solve this problem, we can use the equation P = P0e^(rt), where P is the population at a given time, P0 is the initial population, e is the base of the natural logarithm (approximately 2.71828), r is the growth rate, and t is the time in years.

In this case, the population has doubled in 7 years, so we can write the equation as 2P0 = P0e^(r*7). We need to solve for r.

Dividing both sides of the equation by P0 gives us 2 = e^(7r). Taking the natural logarithm of both sides gives us ln(2) = 7r. Solving for r, we find that r ≈ ln(2)/7.

Now, to determine how long it will take for the population to triple, we need to solve the equation 3P0 = P0e^(r*t) for t. Substituting the value of r we found, we get 3 = e^((ln(2)/7)*t). Taking the natural logarithm of both sides to solve for t, we have ln(3) = (ln(2)/7)*t. Solving for t, we find that t ≈ 7 * ln(3)/ln(2). Rounding the answer to one decimal place, it will take approximately 10.1 years for the population to triple.

It takes approximately 11.1 years for the population to triple.

The population growth can be modeled by the exponential function [tex]P(t)=P_0 \times e^{k t}[/tex], where [tex]P_0[/tex] is the initial population, k is the growth rate, and t is time.

In this case, since the population doubles in 7 years, we have the relationship P(7) = 2 x [tex]P_0[/tex].

[tex]P(7)=P_0 \times e^{7 k}[/tex]

[tex]2 \times P_0=P_0 \times e^{7 k}[/tex]

Divide both sides by [tex]P_0[/tex].

[tex]2=e^{7 k}[/tex]

Take the natural logarithm (ln) of both sides to solve for k:

ln(2) = 7k

[tex]k=\frac{\ln (2)}{7}[/tex]

Now that we have the growth rate k, we can use it to find the time it takes for the population to triple.

[tex]P(t)=P_0 \times e^{k t}[/tex]

For tripling, [tex]P(t)=3 \times P_0[/tex].

[tex]3 \times P_0=P_0 \times e^{k t}[/tex]

[tex]3=e^{k t}[/tex]

Take the natural logarithm again:

ln(3) = kt

[tex]t=\frac{\ln (3)}{k}[/tex]

Substitute the value of k we found earlier:

[tex]t=\frac{\ln (3)}{\frac{\ln (2)}{7}}[/tex]

Using natural logarithm values:

[tex]t \approx \frac{1.0986}{\frac{0.6931}{7}}[/tex]

[tex]t \approx \frac{1.0986}{0.0990}[/tex]

t ≈ 11.09

When the Federal Reserve decreases the reserve rate from 7% to 5%, the amount of money resulting from fraction-reserve banking increases. In this case, an initial deposit of $30,000 would result in $28,500 due to the change in reserve rate.

When the Federal Reserve decreases the reserve rate from 7% to 5%, it means that banks are required to hold a smaller percentage of deposits as reserves. This change allows banks to lend out a larger portion of the initial deposit. In this case, if the initial deposit into a bank is $30,000, the amount that would result from fraction-reserve banking would increase because the bank can now lend out a greater percentage of the deposit.

To calculate the new amount that would result from fraction-reserve banking, we need to calculate the reserve amount and the loan amount.

Therefore, the amount that would result from fraction-reserve banking from an initial deposit of $30,000 would be $28,500.

https://brainly.com/question/30984737

#SPJ2

To graph a linear equation, you need at least two pieces of information: the slope and a point, both intercepts, or two points. One point and the slope, or one intercept and the slope, are also sufficient to graph a linear equation.

For graphing a linear equation, there are several options that provide sufficient information:

To graph a linear equation, you need at least two pieces of information, such as the slope and a point or both intercepts. Having two points is also sufficient since it helps determine the slope. Having just one point or one intercept along with the slope is also enough to graph the line.

The answer is 167,000 because if you round to the 6,782 up it is 7,000. If you round 57,782 up it is 60,000. so then just add those 2 together and plug the 100,000 into it.