Find the length of DF or state what additional information you would need to find it.

Determine all prime numbers a, b and c for which the expression a ^ 2 + b ^ 2 + c ^ 2 - 1 is a perfect square .

kogti [31]

Answer:

The family of all prime numbers such that $a^{2} + b^{2} + c^{2} -1$ is a perfect square is represented by the following solution:

$a$ is an arbitrary prime number. (1)

$b = \sqrt{1 + 2\cdot a \cdot c}$ (2)

$c$ is another arbitrary prime number. (3)

Step-by-step explanation:

From Algebra we know that a second order polynomial is a perfect square if and only if $(x+y)^{2} = x^{2} + 2\cdot x\cdot y + y^{2}$ . From statement, we must fulfill the following identity:

$a^{2} + b^{2} + c^{2} - 1 = x^{2} + 2\cdot x\cdot y + y^{2}$

By Associative and Commutative properties, we can reorganize the expression as follows:

$a^{2} + (b^{2}-1) + c^{2} = x^{2} + 2\cdot x \cdot y + y^{2}$ (1)

Then, we have the following system of equations:

$x = a$ (2)

$(b^{2}-1) = 2\cdot x\cdot y$ (3)

$y = c$ (4)

By (2) and (4) in (3), we have the following expression:

$(b^{2} - 1) = 2\cdot a \cdot c$

$b^{2} = 1 + 2\cdot a \cdot c$

$b = \sqrt{1 + 2\cdot a\cdot c}$

From Number Theory, we remember that a number is prime if and only if is divisible both by 1 and by itself. Then, $a, b, c > 1$ . If $a$ , $b$ and $c$ are prime numbers, then $2\cdot a\cdot c$ must be an even composite number, which means that $a$ and $c$ can be either both odd numbers or a even number and a odd number. In the family of prime numbers, the only even number is 2.

In addition, $b$ must be a natural number, which means that:

$1 + 2\cdot a\cdot c \ge 4$

$2\cdot a \cdot c \ge 3$

$a\cdot c \ge \frac{3}{2}$

But the lowest possible product made by two prime numbers is $2^{2} = 4$ . Hence, $a\cdot c \ge 4$ .

The family of all prime numbers such that $a^{2} + b^{2} + c^{2} -1$ is a perfect square is represented by the following solution:

$a$ is an arbitrary prime number. (1)

$b = \sqrt{1 + 2\cdot a \cdot c}$ (2)

$c$ is another arbitrary prime number. (3)

Example: $a = 2$ , $c = 2$

$b = \sqrt{1 + 2\cdot (2)\cdot (2)}$

$b = 3$

4 0

3 years ago

Taylor uses tape to mark a square play area in the basement for her daughter. The area measures 28 ft2. Is the side length of th

joja [24]

The side length of the square is a rational.

What is the area of the basement?

Any lower storey of a structure that is partially or completely below the lowest contiguous proposed ground level is referred to as a basement floor, provided that the portion of the storey above ground level does not exceed 75 cm.

Given: Taylor uses tape to mark a square play area in the basement for her daughter.

The area measures 28 ft2.

We know that the area of square is side length square.

As the area measures 28 square feet, so the side length of the square is, 28 feet.

Since 28 is a rational number.

Therefore, the side length of the square is a rational.

To know more about the area of basement, click on the link

brainly.com/question/26791387

#SPJ2

4 0

1 year ago

Read 2 more answers

A machine covers 5/8 square foot in 1/4 hour. what is the unit rate?

Ilia_Sergeevich [38]

we know that

Unit Rate is the ratio of two measurements in which the second term is one

so

to find the unit rate divide $(5/8)$ by $(1/4)$

$Unit\ rate=\frac{(5/8)}{(1/4)}= \frac{5}{8}*4= \frac{2.5}{1} =2.5 \frac{ft^{2}}{hour}$

therefore

the answer is

The unit rate is $2.5 \frac{ft^{2}}{hour}$

4 0

3 years ago

Is -1 a solution to 4x - 3 < 5? (look at the picture)

inessss [21]

Answer:

Yes

Step-by-step explanation:

Assuming that the question is asking for -1 to be plugged in as x

4(-1)-3≤5

-4-3≤5

-7≤5

5 0

3 years ago

Determine whether the improper integral converges or diverges, and find the value of each that converges.

kiruha [24]

Answer:

Diverges

Step-by-step explanation:

For the improper integral to converge the limit must exist, if the limit does not exist or is infinite the integral diverges;

$\int\limits^a_b {(2x+4)/x^2+4x+5} \, dx$ #

Where a and be is infinity

The solution of this is infinity and therefore the integral diverges because the first expression is undefined because ∞/∞, the second term is 4·∞ which is infinity and infinity added to any number is infinity

3 0

3 years ago