which choice is a point of discontinuity on the graph of $f(x)=\frac{7}{12x^{2}+2x - 30}$?\n(a) $x =…

which choice is a point of discontinuity on the graph of $f(x)=\frac{7}{12x^{2}+2x - 30}$?\n(a) $x = - 6$\n(b) $x=-\frac{10}{3}$\n(c) $x =-\frac{5}{3}$\n(d) $x=-\frac{3}{2}$

which choice is a point of discontinuity on the graph of $f(x)=\frac{7}{12x^{2}+2x - 30}$?\n(a) $x = - 6$\n(b) $x=-\frac{10}{3}$\n(c) $x =-\frac{5}{3}$\n(d) $x=-\frac{3}{2}$

Answer

Explanation:

Step1: Find where the denominator is zero

Set the denominator $12x^{2}+2x - 30=0$. First, factor out a 2: $2(6x^{2}+x - 15)=0$, then focus on $6x^{2}+x - 15 = 0$.

Step2: Factor the quadratic equation

We need to find two numbers that multiply to $6\times(- 15)=-90$ and add up to 1. The numbers are 10 and - 9. So, $6x^{2}+10x-9x - 15 = 0$. Grouping terms: $2x(3x + 5)-3(3x + 5)=0$, which gives $(3x + 5)(2x-3)=0$.

Step3: Solve for x

Set each factor equal to zero. For $3x+5 = 0$, we have $3x=-5$, so $x =-\frac{5}{3}$. For $2x - 3=0$, we have $2x=3$, so $x=\frac{3}{2}$.

Answer:

C. $x =-\frac{5}{3}$