Understanding Integer Relationships and Quadratic Equations

The video tutorial explains how to solve a problem involving two positive integers where one is four more than twice the other. The difference of their squares is given as 128. The tutorial walks through defining variables, setting up and simplifying the equation, factoring, and finding the solution. The integers are found to be 4 and 12, and the solution is verified by checking the difference of their squares.

What is the relationship between the two positive integers described in the problem?

• One is four more than twice the other.

• One is twice the other.

• One is three times the other.

• One is four less than twice the other.

If x is a positive integer, how can the other integer be expressed?

• x + 4

• 2x + 4

• 4x + 2

• x - 4

What is the equation formed by the difference of the squares of the integers?

• (2x + 4)^2 - x^2 = 128

• (x + 4)^2 - 2x^2 = 128

• (x - 4)^2 - 2x^2 = 128

• (2x - 4)^2 - x^2 = 128

What is the simplified form of the equation after expanding the squares?

• 3x^2 + 16x + 16 = 128

• 4x^2 + 8x + 16 = 128

• 3x^2 + 16x - 112 = 0

• 4x^2 + 8x - 112 = 0

What is the prime factorization of 112 used in the factoring process?

• 2^2 * 28

• 2^3 * 14

• 2^4 * 7

• 2 * 56

Which pair of factors is used to factor the quadratic equation?

• Positive 7 and negative 16

• Negative 4 and positive 28

• Negative 7 and positive 16

• Positive 4 and negative 28

What is the value of x that satisfies the equation?

• x = 3

• x = 6

• x = 4

• x = 5

What are the two positive integers found in the solution?

• 3 and 11

• 4 and 12

• 5 and 13

• 6 and 14

How do you verify the solution is correct?

• By checking the ratio of the integers

• By checking the sum of the integers

• By checking the difference of their squares

• By checking the product of the integers

What is the difference of the squares of the integers 12 and 4?

• 100

• 128

• 144

• 112