Question 113·Medium·Advanced Math

At $x=7$, the denominator $x-7$ is $7-7=0$. Division by zero is undefined, so the original equation requires $x\ne7$.

Type the original equation as $\frac{x_1^2-4x_1-21}{x_1-7}\sim-3$. In this regression, $x_1$ tells Desmos to find a number, and $\sim$ replaces the equals sign. Under PARAMETERS, Desmos shows $x_1=-6$. Since $-6\ne7$, the fraction is defined there, so $x=-6$ satisfies the equation. Choice B.

To factor the numerator is to write it as a product. $-7$ and $3$ multiply to $-21$ and add to $-4$, so the matching factor appears: $x^2-4x-21=(x-7)(x+3)$.

The original denominator is zero at $x=7$, so $x-7$ is nonzero for every allowed input. Cancel that whole factor from the top and bottom: $\frac{(x-7)(x+3)}{x-7}=-3\quad\Longrightarrow\quad x+3=-3,\qquad x\ne7$.

Subtract $3$ from both sides: $x=-6$. This value is allowed by the original denominator because $-6\ne7$, so $x=-6$ satisfies the equation. Choice B.