For which quotient is [tex]$x=7$[/tex] an excluded value?
A. [tex]\frac{x-7}{x^2+4x-21} \div \frac{x^2+49}{x+7}[/tex] B. [tex]\frac{x^2-49}{3x+21} \div \frac{x^2+7x}{3x}[/tex] C. [tex]\frac{x+7}{x^2+6x-7} \div \frac{7}{2x+14}[/tex] D. [tex]\frac{7x}{x^2-10x+21} \div \frac{x+7}{7}[/tex]
To determine for which quotient the value [tex]\( x = 7 \)[/tex] is excluded, we need to check if [tex]\( x = 7 \)[/tex] makes any denominator in the given expressions equal to zero.
### Quotient A [tex]\[
\frac{x-7}{x^2 + 4x - 21} \div \frac{x^2 + 49}{x + 7}
\][/tex] For [tex]\( x = 7 \)[/tex]: - The denominator of the first fraction: [tex]\( x^2 + 4x - 21 \)[/tex] [tex]\[
7^2 + 4 \cdot 7 - 21 = 49 + 28 - 21 = 56 \quad (\text{not zero})
\][/tex] - The denominator of the second fraction: [tex]\( x + 7 \)[/tex] [tex]\[
7 + 7 = 14 \quad (\text{not zero})
\][/tex] Conclusively, [tex]\( x = 7 \)[/tex] does not make any denominator zero in Quotient A.
### Quotient B [tex]\[
\frac{x^2 - 49}{3x + 21} \div \frac{x^2 + 7x}{3x}
\][/tex] For [tex]\( x = 7 \)[/tex]: - The denominator of the first fraction: [tex]\( 3x + 21 \)[/tex] [tex]\[
3 \cdot 7 + 21 = 21 + 21 = 42 \quad (\text{not zero})
\][/tex] - The denominator of the second fraction: [tex]\( 3x \)[/tex] [tex]\[
3 \cdot 7 = 21 \quad (\text{not zero})
\][/tex] Conclusively, [tex]\( x = 7 \)[/tex] does not make any denominator zero in Quotient B.
### Quotient C [tex]\[
\frac{x + 7}{x^2 + 6x - 7} \div \frac{7}{2x + 14}
\][/tex] For [tex]\( x = 7 \)[/tex]: - The denominator of the first fraction: [tex]\( x^2 + 6x - 7 \)[/tex] [tex]\[
7^2 + 6 \cdot 7 - 7 = 49 + 42 - 7 = 84 \quad (\text{not zero})
\][/tex] - The denominator of the second fraction: [tex]\( 2x + 14 \)[/tex] [tex]\[
2 \cdot 7 + 14 = 14 + 14 = 28 \quad (\text{not zero})
\][/tex] Conclusively, [tex]\( x = 7 \)[/tex] does not make any denominator zero in Quotient C.
### Quotient D [tex]\[
\frac{7x}{x^2 - 10x + 21} \div \frac{x + 7}{7}
\][/tex] For [tex]\( x = 7 \)[/tex]: - The denominator of the first fraction: [tex]\( x^2 - 10x + 21 \)[/tex] [tex]\[
7^2 - 10 \cdot 7 + 21 = 49 - 70 + 21 = 0 \quad (\text{zero})
\][/tex] - The denominator of the second fraction: [tex]\( 7 \)[/tex] is a constant and does not depend on [tex]\( x \)[/tex].
Conclusively, [tex]\( x = 7 \)[/tex] makes the denominator of the first fraction zero in Quotient D.
Thus, the correct answer is: [tex]\[ \boxed{4} \][/tex]
