Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Connect with a community of experts ready to provide precise solutions to your questions on our user-friendly Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
To find the difference between the two given expressions:
[tex]\[ \frac{x}{x^2 - 2x - 15} - \frac{4}{x^2 + 2x - 35} \][/tex]
we need to go through a series of steps. Let’s simplify the problem step-by-step.
1. Factorize the Denominators:
First, factorize the denominators of both the fractions.
The denominator [tex]\(x^2 - 2x - 15\)[/tex] factors as:
[tex]\[ x^2 - 2x - 15 = (x - 5)(x + 3) \][/tex]
The denominator [tex]\(x^2 + 2x - 35\)[/tex] factors as:
[tex]\[ x^2 + 2x - 35 = (x + 7)(x - 5) \][/tex]
2. Rewrite the Expressions:
Using these factorizations, we can rewrite each fraction:
[tex]\[ \frac{x}{x^2 - 2x - 15} = \frac{x}{(x - 5)(x + 3)} \][/tex]
[tex]\[ \frac{4}{x^2 + 2x - 35} = \frac{4}{(x + 7)(x - 5)} \][/tex]
3. Common Denominator:
To subtract these two fractions, we need a common denominator. The common denominator for [tex]\((x-5)(x+3)\)[/tex] and [tex]\((x-5)(x+7)\)[/tex] is [tex]\((x-5)(x+3)(x+7)\)[/tex].
4. Rewrite Each Fraction with the Common Denominator:
Rewrite each fraction with the common denominator [tex]\((x - 5)(x + 3)(x + 7)\)[/tex].
For [tex]\(\frac{x}{(x-5)(x+3)}\)[/tex]:
[tex]\[ \frac{x}{(x-5)(x+3)} = \frac{x(x+7)}{(x - 5)(x + 3)(x + 7)} \][/tex]
For [tex]\(\frac{4}{(x+7)(x-5)}\)[/tex]:
[tex]\[ \frac{4}{(x+7)(x-5)} = \frac{4(x+3)}{(x - 5)(x + 3)(x + 7)} \][/tex]
5. Subtract the Fractions:
Now, subtract the two fractions:
[tex]\[ \frac{x(x+7) - 4(x+3)}{(x-5)(x+3)(x+7)} \][/tex]
Simplify the numerator:
[tex]\[ x(x + 7) - 4(x + 3) \][/tex]
[tex]\[ x^2 + 7x - 4x - 12 \][/tex]
[tex]\[ x^2 + 3x - 12 \][/tex]
6. Write the Final Expression:
Putting it all together, the difference is:
[tex]\[ \frac{x^2 + 3x - 12}{(x - 5)(x + 3)(x + 7)} \][/tex]
This is the simplified form of the difference between the given expressions.
Therefore, the difference between:
[tex]\[ \frac{x}{x^2-2x-15} - \frac{4}{x^2+2x-35} \][/tex]
is:
[tex]\[ \frac{x^2 + 3x - 12}{(x-5)(x+3)(x+7)} \][/tex]
[tex]\[ \frac{x}{x^2 - 2x - 15} - \frac{4}{x^2 + 2x - 35} \][/tex]
we need to go through a series of steps. Let’s simplify the problem step-by-step.
1. Factorize the Denominators:
First, factorize the denominators of both the fractions.
The denominator [tex]\(x^2 - 2x - 15\)[/tex] factors as:
[tex]\[ x^2 - 2x - 15 = (x - 5)(x + 3) \][/tex]
The denominator [tex]\(x^2 + 2x - 35\)[/tex] factors as:
[tex]\[ x^2 + 2x - 35 = (x + 7)(x - 5) \][/tex]
2. Rewrite the Expressions:
Using these factorizations, we can rewrite each fraction:
[tex]\[ \frac{x}{x^2 - 2x - 15} = \frac{x}{(x - 5)(x + 3)} \][/tex]
[tex]\[ \frac{4}{x^2 + 2x - 35} = \frac{4}{(x + 7)(x - 5)} \][/tex]
3. Common Denominator:
To subtract these two fractions, we need a common denominator. The common denominator for [tex]\((x-5)(x+3)\)[/tex] and [tex]\((x-5)(x+7)\)[/tex] is [tex]\((x-5)(x+3)(x+7)\)[/tex].
4. Rewrite Each Fraction with the Common Denominator:
Rewrite each fraction with the common denominator [tex]\((x - 5)(x + 3)(x + 7)\)[/tex].
For [tex]\(\frac{x}{(x-5)(x+3)}\)[/tex]:
[tex]\[ \frac{x}{(x-5)(x+3)} = \frac{x(x+7)}{(x - 5)(x + 3)(x + 7)} \][/tex]
For [tex]\(\frac{4}{(x+7)(x-5)}\)[/tex]:
[tex]\[ \frac{4}{(x+7)(x-5)} = \frac{4(x+3)}{(x - 5)(x + 3)(x + 7)} \][/tex]
5. Subtract the Fractions:
Now, subtract the two fractions:
[tex]\[ \frac{x(x+7) - 4(x+3)}{(x-5)(x+3)(x+7)} \][/tex]
Simplify the numerator:
[tex]\[ x(x + 7) - 4(x + 3) \][/tex]
[tex]\[ x^2 + 7x - 4x - 12 \][/tex]
[tex]\[ x^2 + 3x - 12 \][/tex]
6. Write the Final Expression:
Putting it all together, the difference is:
[tex]\[ \frac{x^2 + 3x - 12}{(x - 5)(x + 3)(x + 7)} \][/tex]
This is the simplified form of the difference between the given expressions.
Therefore, the difference between:
[tex]\[ \frac{x}{x^2-2x-15} - \frac{4}{x^2+2x-35} \][/tex]
is:
[tex]\[ \frac{x^2 + 3x - 12}{(x-5)(x+3)(x+7)} \][/tex]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
