Reduce answers to lowest terms.
1.) 2x/2x^2+5x-3 + 5/x^2-9 =
2.) x-3/12x^3-18x^2 - x+1/16x^2-24x =
3.) 2x/x^2-2x-15 - x-2/3x^2+9x =
Use parentheses to isolate numerators and denominators. For example,
3.) 2x/(x^2-2x-15) - (x-2)/(3x^2+9x)
= 2x/[(x-5)(x+3)] + (x-2)/[3x(x+3)]
= [1/(x+3)][2x/(x-5) + (x-2)/(3x)]
2/a-b+3/a+b
To reduce the given expressions to their lowest terms, we need to simplify the fractions as much as possible by factoring and canceling out common factors.
1.) 2x/(2x^2+5x-3) + 5/(x^2-9)
To simplify the first fraction, let's factor the denominator of the first fraction (2x^2+5x-3):
The factors of 2x^2+5x-3 are (2x-1)(x+3)
So, the expression becomes:
2x/(2x-1)(x+3) + 5/(x^2-9)
Now, let's factor the denominator of the second fraction (x^2-9):
The factors of x^2-9 are (x+3)(x-3)
The expression is now:
2x/(2x-1)(x+3) + 5/(x+3)(x-3)
Since we have a common denominator (2x-1)(x+3)(x-3), we can add the fractions:
[2x(x-3) + 5] / (2x-1)(x+3)(x-3)
Simplify the numerator:
[2x^2 - 6x + 5] / (2x-1)(x+3)(x-3)
The expression is now in its lowest terms.
2.) (x-3)/(12x^3 - 18x^2) - (x+1)/(16x^2 - 24x)
To simplify fractions in the expression, let's factor the denominators:
12x^3 - 18x^2 = 6x^2(2x - 3)
16x^2 - 24x = 8x(2x - 3)
The expression becomes:
(x-3)/(6x^2(2x - 3)) - (x+1)/(8x(2x - 3))
Since we have a common denominator, we can subtract the fractions:
[(x-3)8x - (x+1)6x^2] / 6x^2(2x - 3)
Simplify the numerator:
(8x^2 - 24x - 6x^2 + 18x - 6) / 6x^2(2x - 3)
Combine like terms:
(2x^2 - 6x - 6) / 6x^2(2x - 3)
Now, let's factor the numerator:
2(x^2 - 3x - 3) / 6x^2(2x - 3)
Reduce the fraction by canceling common factors:
(x^2 - 3x - 3) / 3x^2(2x - 3)
The expression is now in its lowest terms.
3.) 2x/(x^2 - 2x - 15) - (x-2)/(3x^2 + 9x)
To simplify fractions in the expression, let's factor the denominators:
x^2 - 2x - 15 = (x - 5)(x + 3)
3x^2 + 9x = 3x(x + 3)
The expression becomes:
2x/[(x - 5)(x + 3)] - (x - 2)/[3x(x + 3)]
Since we have a common denominator, we can subtract the fractions:
[2x(3x) - (x - 2)(x - 5)] / (x - 5)(x + 3)[3x]
Simplify the numerator:
[6x^2 - (x^2 - 7x + 10)] / (x - 5)(x + 3)[3x]
Combine like terms:
[6x^2 - x^2 + 7x - 10] / (x - 5)(x + 3)[3x]
Simplify the numerator:
[5x^2 + 7x - 10] / (x - 5)(x + 3)[3x]
Now, let's factor the numerator:
(5x + 10)(x - 1) / (x - 5)(x + 3)[3x]
Reduce the fraction by canceling common factors:
(5(x + 2)(x - 1)) / (x - 5)(x + 3)[3x]
The expression is now in its lowest terms.