Question
whatare the staps to solving this equasion
x^2 -5x /x^2 +3x -10 / x^2 -25 / x^2 - 8x + 12
x^2 -5x /x^2 +3x -10 / x^2 -25 / x^2 - 8x + 12
Answers
GPT-4o mini
To solve the expression given by
\[
\frac{x^2 - 5x}{x^2 + 3x - 10} \div \frac{x^2 - 25}{x^2 - 8x + 12},
\]
you can follow these steps:
1. **Rewrite the Division as Multiplication**:
In mathematics, dividing by a fraction is the same as multiplying by its reciprocal. So, we can rewrite the expression as:
\[
\frac{x^2 - 5x}{x^2 + 3x - 10} \times \frac{x^2 - 8x + 12}{x^2 - 25}.
\]
2. **Factor Each Polynomial**:
Next, we need to factor each polynomial if possible.
- **For \(x^2 - 5x\)**:
\[
x^2 - 5x = x(x - 5).
\]
- **For \(x^2 + 3x - 10\)**:
To factor this, we look for two numbers that multiply to \(-10\) and add to \(3\), which are \(5\) and \(-2\).
\[
x^2 + 3x - 10 = (x + 5)(x - 2).
\]
- **For \(x^2 - 25\)** (this is a difference of squares):
\[
x^2 - 25 = (x - 5)(x + 5).
\]
- **For \(x^2 - 8x + 12\)**:
We look for two numbers that multiply to \(12\) and add to \(-8\), which are \(-6\) and \(-2\).
\[
x^2 - 8x + 12 = (x - 6)(x - 2).
\]
3. **Substituting Factored Forms**:
Replace the original expression with the factored forms:
\[
\frac{x(x - 5)}{(x + 5)(x - 2)} \times \frac{(x - 6)(x - 2)}{(x - 5)(x + 5)}.
\]
4. **Cancel Common Factors**:
Look for factors that appear in both the numerator and the denominator:
- \(x - 5\) cancels out,
- \(x + 5\) cancels out,
- \(x - 2\) cancels out.
After canceling, the expression simplifies to:
\[
\frac{x(x - 6)}{1} = x(x - 6).
\]
5. **Final Expression**:
The simplified result is:
\[
x(x - 6) = x^2 - 6x.
\]
6. **Determine Restrictions**:
Remember to state restrictions based on the original denominators:
- From \(x^2 + 3x - 10\): \(x \neq -5\) and \(x \neq 2\),
- From \(x^2 - 25\): \(x \neq 5\) and \(x \neq -5\),
- From \(x^2 - 8x + 12\): \(x \neq 6\) and \(x \neq 2\).
**Final Answer**: The simplified expression is \(x^2 - 6x\) with restrictions \(x \neq -5, 2, 5, 6\).
\[
\frac{x^2 - 5x}{x^2 + 3x - 10} \div \frac{x^2 - 25}{x^2 - 8x + 12},
\]
you can follow these steps:
1. **Rewrite the Division as Multiplication**:
In mathematics, dividing by a fraction is the same as multiplying by its reciprocal. So, we can rewrite the expression as:
\[
\frac{x^2 - 5x}{x^2 + 3x - 10} \times \frac{x^2 - 8x + 12}{x^2 - 25}.
\]
2. **Factor Each Polynomial**:
Next, we need to factor each polynomial if possible.
- **For \(x^2 - 5x\)**:
\[
x^2 - 5x = x(x - 5).
\]
- **For \(x^2 + 3x - 10\)**:
To factor this, we look for two numbers that multiply to \(-10\) and add to \(3\), which are \(5\) and \(-2\).
\[
x^2 + 3x - 10 = (x + 5)(x - 2).
\]
- **For \(x^2 - 25\)** (this is a difference of squares):
\[
x^2 - 25 = (x - 5)(x + 5).
\]
- **For \(x^2 - 8x + 12\)**:
We look for two numbers that multiply to \(12\) and add to \(-8\), which are \(-6\) and \(-2\).
\[
x^2 - 8x + 12 = (x - 6)(x - 2).
\]
3. **Substituting Factored Forms**:
Replace the original expression with the factored forms:
\[
\frac{x(x - 5)}{(x + 5)(x - 2)} \times \frac{(x - 6)(x - 2)}{(x - 5)(x + 5)}.
\]
4. **Cancel Common Factors**:
Look for factors that appear in both the numerator and the denominator:
- \(x - 5\) cancels out,
- \(x + 5\) cancels out,
- \(x - 2\) cancels out.
After canceling, the expression simplifies to:
\[
\frac{x(x - 6)}{1} = x(x - 6).
\]
5. **Final Expression**:
The simplified result is:
\[
x(x - 6) = x^2 - 6x.
\]
6. **Determine Restrictions**:
Remember to state restrictions based on the original denominators:
- From \(x^2 + 3x - 10\): \(x \neq -5\) and \(x \neq 2\),
- From \(x^2 - 25\): \(x \neq 5\) and \(x \neq -5\),
- From \(x^2 - 8x + 12\): \(x \neq 6\) and \(x \neq 2\).
**Final Answer**: The simplified expression is \(x^2 - 6x\) with restrictions \(x \neq -5, 2, 5, 6\).