Algebra

(2x^2 - 5x - 3) / (x - 3) = 4

Step-by-step solution with explanation

Final Answer

x = 3 is excluded (undefined); no valid solution exists

Step-by-step solution

1

Note the restriction on x

The denominator is (x - 3). If x = 3, we'd be dividing by zero, which is undefined. So x = 3 is not allowed.
2

Multiply both sides by (x - 3)

We clear the fraction by multiplying both sides by (x - 3). This is valid as long as x ≠ 3.
3

Expand the right side

Distribute the 4 across (x - 3) to get 4x - 12.
4

Move all terms to one side

Subtract 4x and add 12 to both sides so everything is on the left side.
5

Factor the quadratic

We look for two factors of 2x² - 9x + 9. Checking: (2x - 3)(x - 3) = 2x² - 6x - 3x + 9 = 2x² - 9x + 9. ✓
6

Solve each factor for x

Set each factor equal to zero and solve. This gives two candidate solutions: x = 3/2 and x = 3.
7

Reject x = 3 due to restriction

We already said x ≠ 3, so that answer is thrown out. Now check x = 3/2 in the original equation: (2(9/4) - 5(3/2) - 3) / (3/2 - 3) = (9/2 - 15/2 - 6/2) / (-3/2) = (-12/2) / (-3/2) = (-6) / (-3/2) = 4. ✓

Understanding this problem

Learning Insight

When you multiply both sides of an equation by an expression containing x, you might accidentally create extra 'extraneous' solutions. That's why we always check candidates against any restrictions found at the start. The algebra is valid, but the original equation sets limits on what x can be.

Quick Tip

Always write down your domain restrictions FIRST (values that make the denominator zero), then solve, then cross off any answer that hits those restricted values.

Common Mistake

Students forget to check solutions against the domain restriction and happily write x = 3 as a valid answer — but plugging it in causes division by zero, making the equation undefined.