Question 1015·Medium·Math

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What value of $x$ satisfies the equation

\frac{x^2 - 4x - 21}{x - 7} = -3

A rational equation has a variable in a denominator, so solving it also means checking where the fraction is defined. Rule out any denominator zero first: an input excluded by the original fraction stays excluded, even if you cancel a factor. A Desmos regression can solve the original equation in one line. The tempting slip is to cancel first and forget the excluded input.

Hints

Hint 1
In a rational equation, division by zero is undefined. Which value makes the original denominator $x-7$ equal $0$ ? Keep that value out, even if the denominator later disappears from your work.
Hint 2
A regression asks Desmos to find an unknown number. Replace every $x$ with $x_1$ and the equals sign with $\sim$ , then read the value under PARAMETERS.
Hint 3
Before choosing, compare the value Desmos finds with the excluded input. A number solves the original equation only if the fraction is defined there.

Step-by-step

Approach 1: Solve the original equation in Desmos

Step 1
Exclude the denominator zero
At $x=7$ , the denominator $x-7$ is $7-7=0$ . Division by zero is undefined, so the original equation requires $x\ne7$ .
Step 2
Find and check the solution
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.

Approach 2: Factor to see why it works

Step 1
Find the matching factor
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: $x^2-4x-21=(x-7)(x+3)$ .
Step 2
Cancel without losing the restriction
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: $\frac{(x-7)(x+3)}{x-7}=-3\quad\Longrightarrow\quad x+3=-3,\qquad x\ne7$ .
Step 3
Solve the shorter equation
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.

Question 1015·Medium·Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this