Problem 128 - College Algebra

Question

Solve each equation by the method of your choice. $$\frac{x-1}{x-2}+\frac{x}{x-3}=\frac{1}{x^{2}-5 x+6}$$

Step-by-Step Solution

Answer

The solutions are \(x = 1.5 + 0.5\sqrt{5}\) and \(x = 1.5 - 0.5\sqrt{5}\).

Step 1: Factor the denominator

Factor the denominator \(x^{2}-5x+6\) into \((x-2)(x-3)\). Now, you can see it becomes the common denominator for all fractions in the equations.

Step 2: Multiply every term with common denominator

Multiply the entire equation by the common denominator \((x-2)(x-3)\). The equation would simplify to \[(x-1)(x-3) + x(x-2) = 1.\] Now we have a quadratic equation.

Step 3: Simplify and rewrite the equation

Simplify the expression by expanding and combining like terms: \[x^2 - 4x + 3 + x^2 - 2x = 1\], this simplifies to: \(2x^2 - 6x + 3 = 1\).

Step 4: Solve the quadratic equation

To solve the quadratic equation, subtract 1 from both sides to set the equation to zero: \(2x^2 - 6x + 2 = 0\). Divide all terms by 2 and the equation becomes \(x^2 - 3x + 1 = 0\). Find the roots by applying the quadratic formula: \(x = [-b \pm \sqrt{(b^2 - 4ac)}] / 2a\), where \(a=1\), \(b=-3\), and \(c=1\).