Is 4√5 rational or irrational

The solution of the equation found by completing the square are;

x = 7 - √(53) and 7 + √(53)

Completing the square method involves writing the quadratic expression of the form a·x² + b·x + c as (x + d)²  + e, where d and e are constants.

Moving the constant to one side of the equation, we get;

x² - 14·x = 4

We complete the square on the left side of the equation by adding and subtracting the square of the half the coefficient of x term inside the parenthesis. Therefore, we add and subtract (-14/2)² = 49, as follows;

(x² - 14·x + 49) - 49 = 4

Therefore; (x - 7)² - 49 = 4

The -49 is moved to the right side of the equation to get;

(x - 7)² = 4 + 49 = 53

Taking the square root of both sides, we get;

x - 7 = ± √(53)

Therefore; x = 7 ± √(53)

The solution is; x = 7 + √(53) and 7 - √(53)

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Answer:

Option A -distributive property

Step-by-step explanation:

Given : expression     -3 + 2(2 + x + 3)

Step 1: = -3 + 2(x + 2 + 3)

Step 2: = -3 + 2(x + 5)

Step 3: = -3 + 2x + 10

Step 4: = 2x + (-3) + 10

Step 5: = 2x + 7

In step 3 distributive property is applied i.e. c(a+b)=ca+cb

in step 2- 2(x+5) when we apply distributive property it became

2x+10

Therefore, option A is correct

Answer:

The value of y is 7.

Step-by-step explanation:

The given equation is

[tex]log_{11}(y+8)+log_{11}4=log_{11}60[/tex]

[tex]log_{11}[(y+8)4]=log_{11}60[/tex]               [tex][\because log_ax+log_ay=log_axy][/tex]

[tex]log_{11}(4y+32)=log_{11}60[/tex]

Equating both sides.

[tex]4y+32=60[/tex]

[tex]4y=60-32[/tex]

[tex]4y=28[/tex]

Divide both sides by 4.

[tex]y=7[/tex]

Therefore the value of y is 7.

Answer:

the answer is b) 70.7 feet

Step-by-step explanation:

thank you for amazing answers on the board.

Final answer:

Bobby can buy 5 cups of coffee at $2.25 each and 3 cups of hot chocolate at $1.50 each to meet his budget of $15.75 for 8 cups in total. We find this by solving a system of equations.

Explanation:

Bobby wants to buy a total of 8 cups of coffee and hot chocolate for his co-workers with a budget of $15.75. Since each cup of coffee costs $2.25 and each cup of hot chocolate costs $1.50, we need to find out how many cups of each he can buy.

Let's denote the number of cups of coffee as C and the number of cups of hot chocolate as H.

The total cost for coffee will be $2.25 × C and for hot chocolate $1.50 × H.

The total amount spent is $15.75, and the total number of cups is 8, so we have two equations: 2.25C + 1.50H = 15.75 and C + H = 8.

We can solve this system of equations using substitution or elimination. For simplicity, we'll use substitution.

From C + H = 8, we can express H in terms of C: H = 8 - C.

Substitute H in the cost equation: 2.25C + 1.50(8 - C) = 15.75.

Simplify and solve for C: 2.25C + 12 - 1.50C = 15.75, which simplifies to 0.75C = 3.75, and therefore C = 5.

Now, plug C = 5 into H = 8 - C to find H: H = 3.

Bobby can buy 5 cups of coffee and 3 cups of hot chocolate.

The percent of increase in the number of women motorcyclists over the 8-year period is approximately 45.77%, calculated by finding the difference between the final and initial number of motorcyclists, dividing by the initial number, and converting the ratio to a percentage.

To calculate the percent of increase in the number of women motorcyclists over the 8-year period provided, follow these steps:

Determine the initial and final values. Initial value: 437,000; Final value: 637,000.

Calculate the increase by subtracting the initial value from the final value: 637,000 - 437,000 = 200,000.

Divide the increase by the initial value to find the increase ratio: 200,000 / 437,000 ≈ 0.4577.

Multiply the increase ratio by 100 to convert it to a percentage: 0.4577 * 100 ≈ 45.77%.

Therefore, the percent of increase in the number of women motorcyclists during these 8 years is approximately 45.77%.

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Answer:

The answer is C. 75 hours.

Step-by-step explanation:

In order to determine the hours, we have to know about direct proportionality.

The direct proportionality happens when two variables increase at the same time and their rate is a constant. It means that if one variable is increased, so the other variable will be increased too.

I have attached an image that shows a graph of two variables with direct proportionality.

In this case, we can get the constant of both variables: losing weight and the spending time.

[tex]k=\frac{Weight}{Time}=\frac{1}{3} =0.33[/tex]

Then we need to know how many hours Sam will spend to lose 25 lbs:

[tex]\frac{25}{T}=k=0.33\\ T=\frac{25}{0.33}\\ T=75.76[/tex] hours.

Finally, Sam needs to spend 75 hours to lose 25 lbs.

The inequality would be equal to n  ≥ 7.

If the equation or inequality contains variable terms, then there might be some values of those variables for which that equation or inequality might be true. Such values are called solution to that equation or inequality. Set of such values is called solution set to the considered equation or inequality.

We are given that the phrase as;

"A number n is at least 7"

Therefore, the inequality would be;

n  ≥ 7

Or

n = 7

Thus, the inequality would be equal to n  ≥ 7.

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Answer:

commutative property of addition

Step-by-step explanation:

Answer:

[tex]\frac{1}{1} \times \frac{3}{5}[/tex]

Step-by-step explanation:

The complete question is

Yen calculates the product 5/8 x 24/25. before he multiplies, he simplifies all the factors. what does the problem look like after he simplifies the factors?

The given product is

[tex]\frac{5}{8} \times \frac{24}{25}[/tex]

Then, Yen simplifies and the problem looks like

[tex]\frac{1}{1} \times \frac{3}{5}[/tex]

Because, we simplified 5 with 25 and 8 with 24.

Therefore, the result of the product is

[tex]\frac{3}{5}[/tex]

The ratio 60 cents: $2 simplifies to 3:10 after converting dollars to cents and dividing both terms by the greatest common divisor, which is 20.

To simplify the ratio 60 cents: $2, we need to first express both quantities in the same units. Since 1 dollar is equal to 100 cents, $2 is equivalent to 200 cents. Thus, the ratio becomes 60 cents : 200 cents.

Next, we simplify the ratio by dividing both sides by the greatest common divisor. In this case, both 60 and 200 are divisible by 20, which is the greatest common divisor. So we divide both sides by 20:

[tex]\frac{60}{20}[/tex] = 3

[tex]\frac{200}{20}[/tex] = 10

Therefore, the simplified ratio is 3:10.