Algebra

(x^2 - 9)/(x + 3) + 2 = x

Step-by-step solution with explanation

Final Answer

No solution

Step-by-step solution

1

Factor the numerator completely

The numerator is a difference of squares. We factor it as two binomials: (x+3)(x-3). This sets us up to simplify the fraction.
2

Simplify the fraction, note restriction

The (x+3) on top and bottom cancel, leaving x-3. We must remember x ≠ -3 because dividing by zero is undefined.
3

Rewrite the equation after simplifying

Replace the fraction with x-3. Now we have a simple linear equation to solve.
4

Combine like terms on the left

-3 + 2 = -1, so the left side becomes x - 1.
5

Subtract x from both sides

When we subtract x from both sides, the x terms cancel completely. We get -1 = 0, which is never true.
6

Conclude there is no solution

A false statement like -1 = 0 means there is no value of x that satisfies the original equation. The equation has no solution.

Understanding this problem

Learning Insight

When you simplify a rational expression by canceling a factor, you lose a potential solution at the restricted value (here x = -3). In this problem, the simplification actually reduces the equation to an identity that is always false, meaning the two sides of the equation represent parallel lines — they never intersect.

Quick Tip

Whenever you cancel a factor from a fraction, immediately write down the restriction (e.g., x ≠ -3). Then if you get a contradiction like -1 = 0, you know right away there's no solution.

Common Mistake

Students often cancel (x+3) and then forget the restriction x ≠ -3, or they assume that because the algebra 'worked out' there must be a solution. A contradiction (-1 = 0) always means no solution, no matter how clean the steps looked before it.