How do you show 2x-3 over x-3 - 2x-1 over 2x plus 1 can be 2xsquared plus 3x - 6 over bracketx-3bracket bracket2x plus 1bracket?

Answer 1

Show that [(2x-3)/(x-3)] - [(2x-1)/(2x+1)] = (2x^2+3x-6)/(x-3)(2x+1)

First, you want to get a common denominator. The easiest way to find a common denominator for these two terms is to multiply the denominators together.

(x-3)*(2x-1) You do not need to multiply them out because it is not done so in the answer.

So, in the first term the denominator was multiplied by 2x+1. Since the denominator was multiplied by 2x+1, the numerator must also be multiplied by 2x+1 to ensure that the value has not changed. Its much like multiplying by 1/1, but instead its (2x+1)/(2x+1)

The numerator will be (2x-3)*(2x+1). Multiplying these together will result in 4x2-6x+2x-3. Adding like terms give us 4x2-4x-3

So the first term becomes (4x2-4x-3)/(x-3)(2x+1)

For the second term we multiply the numerator by x-3 instead

(x-3)*(2x-1) The result is (2x2-6x-x+3). Add like terms to give (2x2-7x+3)

The second term becomes (2x2-7x+3)/(x-3)(2x+1)

Since the two terms have the same denominator we only have to subtract the numerators from each other to find the answer.

(4x2-4x-3)-(2x2-7x+3) Now subtract like terms from each other, but pay attention to the sign of each term.

4x2 - 2x2 = 2x2

(-4x) - (-7x) = 3x

(-3) - 3 = (-6)

Combine those terms and the result is 2x2+3x-6

This the numerator, so bringing back the denominator will give us the final answer.

(2x2+3x-6)/(x-3)(2x+1)