Question
Rational Expressions & Equations Unit Test
8 of 208 of 20 Items
Question
Express in simplest form: 5x2+10xx2+7x+12÷x2−4x2+x−6 .(1 point)
Responses
5x+4
Start Fraction 5 over x plus 4 End Fraction
5xx+4
Start Fraction 5 x over x plus 4 End Fraction
x+45x
Start Fraction x plus 4 over 5 x End Fraction
54
8 of 208 of 20 Items
Question
Express in simplest form: 5x2+10xx2+7x+12÷x2−4x2+x−6 .(1 point)
Responses
5x+4
Start Fraction 5 over x plus 4 End Fraction
5xx+4
Start Fraction 5 x over x plus 4 End Fraction
x+45x
Start Fraction x plus 4 over 5 x End Fraction
54
Answers
GPT-4o mini
To simplify the expression \( \frac{5x^2 + 10x^2 + 7x + 12}{x^2 - 4x^2 + x - 6} \), we first need to correct the denominator because the expression given seems off.
Let's clarify the expression and simplify each part:
### Step 1: Combine Like Terms
The numerator is:
\[ 5x^2 + 10x^2 + 7x + 12 = 15x^2 + 7x + 12 \]
The denominator can be simplified. Assuming the expression was intended as \( x^2 - 4x + x - 6 \) and combining the like terms yields:
\[ (x^2 + x - 4x - 6) = (x^2 - 3x - 6) \]
So now our expression is:
\[ \frac{15x^2 + 7x + 12}{x^2 - 3x - 6} \]
### Step 2: Factor the Numerator and Denominator
#### To factor the numerator \( 15x^2 + 7x + 12 \):
We look for two numbers that multiply to \( 15 \times 12 = 180 \) and add to \( 7 \). The pair \( 15 \) and \( 12 \) doesn't seem to help easily.
Let's just use the quadratic formula to find its factors.
The roots will be:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 15 \cdot 12}}{2 \cdot 15}
\]
Calculating this gives complex factors, so we will leave it as is.
#### To factor the denominator \( x^2 - 3x - 6 \):
We can look for factors. This factors into:
\((x - 6)(x + 1)\).
### Final Expression:
Since we couldn't factor the numerator neatly, we are left with:
\[
\frac{15x^2 + 7x + 12}{(x - 6)(x + 1)}
\]
Unfortunately, we cannot simplify further without specific values for x.
Therefore among the responses given and based on workings, the correct answer from your options would be:
1. **None of the provided responses seem accurate regarding the function calculated above.**
To summarize: **Check the expressions given for correctness before choosing based on factors.**
Let's clarify the expression and simplify each part:
### Step 1: Combine Like Terms
The numerator is:
\[ 5x^2 + 10x^2 + 7x + 12 = 15x^2 + 7x + 12 \]
The denominator can be simplified. Assuming the expression was intended as \( x^2 - 4x + x - 6 \) and combining the like terms yields:
\[ (x^2 + x - 4x - 6) = (x^2 - 3x - 6) \]
So now our expression is:
\[ \frac{15x^2 + 7x + 12}{x^2 - 3x - 6} \]
### Step 2: Factor the Numerator and Denominator
#### To factor the numerator \( 15x^2 + 7x + 12 \):
We look for two numbers that multiply to \( 15 \times 12 = 180 \) and add to \( 7 \). The pair \( 15 \) and \( 12 \) doesn't seem to help easily.
Let's just use the quadratic formula to find its factors.
The roots will be:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 15 \cdot 12}}{2 \cdot 15}
\]
Calculating this gives complex factors, so we will leave it as is.
#### To factor the denominator \( x^2 - 3x - 6 \):
We can look for factors. This factors into:
\((x - 6)(x + 1)\).
### Final Expression:
Since we couldn't factor the numerator neatly, we are left with:
\[
\frac{15x^2 + 7x + 12}{(x - 6)(x + 1)}
\]
Unfortunately, we cannot simplify further without specific values for x.
Therefore among the responses given and based on workings, the correct answer from your options would be:
1. **None of the provided responses seem accurate regarding the function calculated above.**
To summarize: **Check the expressions given for correctness before choosing based on factors.**