How do i Solve this. It says perform indicated operations and express in lowest terms.
5x 20
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x^2-5x+6 x^2-6x+8
is it 5x or 5 times 20
To solve this problem, you need to perform the indicated operations and express the result in lowest terms.
Step 1: Find the least common denominator (LCD)
The LCD is the least common multiple of the denominators (x^2 - 5x + 6) and (x^2 - 6x + 8). To find the LCD, we need to factorize the denominators and find their common factors.
(x^2 - 5x + 6) factors into (x - 2)(x - 3)
(x^2 - 6x + 8) factors into (x - 2)(x - 4)
The common factors are (x - 2). So the LCD is (x - 2)(x - 3)(x - 4).
Step 2: Rewrite the fractions with the LCD
Multiply the numerator and denominator of each fraction by the missing factors of the LCD to obtain the equivalent fractions with the LCD as the denominator.
For the first fraction, multiply the numerator and denominator by (x - 3)(x - 4):
5x * (x - 3)(x - 4) / (x^2 - 5x + 6)(x - 3)(x - 4)
For the second fraction, multiply the numerator and denominator by (x - 2)(x - 3):
20 * (x - 2)(x - 3) / (x^2 - 6x + 8)(x - 2)(x - 3)
Step 3: Combine the fractions
Now that both fractions have the same denominator, you can add them together. Simply add the numerators while keeping the denominator.
(5x(x - 3)(x - 4) + 20(x - 2)(x - 3)) / [(x^2 - 5x + 6)(x - 3)(x - 4)]
Step 4: Simplify the expression
Expand and combine like terms in the numerator, and leave the factors in the denominator as they are.
(5x^3 - 35x^2 + 72x - 120 + 20x^2 - 60x) / [(x^2 - 5x + 6)(x - 3)(x - 4)]
Simplifying further, we get:
(5x^3 - 15x^2 + 12x - 120) / [(x^2 - 5x + 6)(x - 3)(x - 4)]
This is the simplified form of the expression in lowest terms after performing the indicated operations.