Which of the following expressions can be written as (ax+b)(ax-b), where a and b are integers? Select all that apply. A. x^2- 9 B. x^2-11 C. 4x^2-1 D. 4x^2-2 E. 9x^2-4
AAAaannndd the bot gets it wrong yet again!
what about 4x^2-1 = (2x+1)(2x-1) ?
To see if an expression can be written as (ax+b)(ax-b), we need to look for factorable quadratic expressions.
Let's expand (ax+b)(ax-b) to see what it looks like:
(ax+b)(ax-b) = a^2x^2 - abx + abx - b^2
= a^2x^2 - b^2
Now let's examine each expression:
A. x^2 - 9:
This can be factored as (x-3)(x+3), not in the form (ax+b)(ax-b).
B. x^2 - 11:
This cannot be factored further, so it is not in the form (ax+b)(ax-b).
C. 4x^2 - 1:
This cannot be factored further, so it is not in the form (ax+b)(ax-b).
D. 4x^2 - 2:
This cannot be factored further, so it is not in the form (ax+b)(ax-b).
E. 9x^2 - 4:
This can be factored as (3x-2)(3x+2), which is in the form (ax+b)(ax-b).
Therefore, the expression E. 9x^2 - 4 can be written as (ax+b)(ax-b), where a and b are integers.
To determine which of the given expressions can be written as (ax+b)(ax-b), we need to check if the expression is a difference of two perfect squares.
The general form of a difference of squares is (a^2 - b^2) = (a + b)(a - b), where a and b are integers.
Let's examine each expression:
A. x^2 - 9:
This expression can be factored as (x + 3)(x - 3), which is not in the form (ax + b)(ax - b). Therefore, A is not the correct answer.
B. x^2 - 11:
This expression cannot be factored into the form (ax + b)(ax - b), as it is not a difference of perfect squares. Therefore, B is not the correct answer.
C. 4x^2 - 1:
This expression is not a difference of two perfect squares, as 1 is not a perfect square. Therefore, C is not the correct answer.
D. 4x^2 - 2:
This expression is not a difference of two perfect squares, as 2 is not a perfect square. Therefore, D is not the correct answer.
E. 9x^2 - 4:
This expression can be factored as (3x + 2)(3x - 2), which is in the form (ax + b)(ax - b). Therefore, E is the correct answer.
In conclusion, the expression 9x^2 - 4 can be written as (ax + b)(ax - b), where a and b are integers.