for which integers will the square of the integer decreased by twice the integer produce a difference of less than 4? please answer and explain how you got the answer...please????!!
how did you get that ms sue?
Trial and error -- but it didn't take long.
2^2 = 4
4 - 4 = 0 >> Nope. It's not 2.
3^2 = 9
9 - 6 = 3 >> Yep! That works.
Obviously 4 is not the answer either.
To solve this problem, we need to translate the given sentence into an equation and then solve it.
Let's break down the problem step by step:
1. Let's assume the integer as "x."
2. The square of the integer is x^2.
3. The given sentence states that the square of the integer decreased by twice the integer produces a difference of less than 4, which can be represented as:
x^2 - 2x < 4
Now, let's solve this inequality to find the range of integers that satisfy the given condition.
1. Subtract 4 from both sides of the inequality:
x^2 - 2x - 4 < 0
2. We can rewrite this quadratic inequality as:
(x - 2)(x + 2) < 0
3. Now, let's consider the intervals of x where the expression (x - 2)(x + 2) is less than 0. To do this, we can set up a number line and mark the critical points -2 and 2:
---(-2)---2---(+2)---
4. We need to determine the sign of the expression (x - 2)(x + 2) in each interval:
- In the interval (-∞, -2), both factors are negative, so the product is positive.
- In the interval (-2, 2), the factor (x - 2) is negative, while the factor (x + 2) is positive, so the product is negative.
- In the interval (2, +∞), both factors are positive, so the product is positive.
5. Since we want the expression (x - 2)(x + 2) to be less than zero, the solution is the interval where the expression is negative, which is (-2, 2).
Therefore, the range of integers that satisfy the given condition is from -1 to 1, excluding -2 and 2.
To summarize: for the integers -1, 0, and 1, the square of the integer decreased by twice the integer produces a difference less than 4.