%2F(x---2)---(1---x)%2Fx-%3D-17%2F4-x-ne0-2.jpeg "(x + 3)/(x - 2) - (1 - X)/x = 17/4 X Ne0 2")
Solving an Algebraic Equation
In this article, we will solve the algebraic equation:
$\frac{x + 3}{x - 2} - \frac{1 - x}{x} = \frac{17}{4},$
where $x \neq 0$ and $x \neq 2$.
Step 1: Simplify the equation
To start, we can simplify the equation by combining the fractions on the left-hand side:
$\frac{x + 3}{x - 2} - \frac{1 - x}{x} = \frac{x^2 + 3x - x + 3}{x(x - 2)} - \frac{x - 1}{x}$
$= \frac{x^2 + 2x + 3}{x(x - 2)} - \frac{x - 1}{x}$
Step 2: Get a common denominator
Next, we need to find a common denominator for the two fractions on the left-hand side. The least common multiple of $x$ and $x - 2$ is $x(x - 2)$. So, we can rewrite the equation as:
$\frac{x^2 + 2x + 3}{x(x - 2)} - \frac{(x - 1)(x - 2)}{x(x - 2)} = \frac{17}{4}$
Step 3: Simplify the numerator
Now, we can simplify the numerator:
$\frac{x^2 + 2x + 3 - (x^2 - 3x + 2)}{x(x - 2)} = \frac{17}{4}$
$\frac{5x + 1}{x(x - 2)} = \frac{17}{4}$
Step 4: Cross-multiply
Next, we can cross-multiply to get rid of the fraction:
$4(5x + 1) = 17x(x - 2)$
Step 5: Expand and simplify
Now, we can expand and simplify the equation:
$20x + 4 = 17x^2 - 34x$
$17x^2 - 54x + 4 = 0$
Step 6: Solve the quadratic equation
Finally, we can solve the quadratic equation using the quadratic formula:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
In this case, $a = 17, b = -54$, and $c = 4$. Plugging these values into the formula, we get:
$x = \frac{54 \pm \sqrt{2916 - 272}}{34}$
Simplifying, we get two possible values for $x$:
$x = \frac{54 \pm \sqrt{2644}}{34}$
$x \approx 2.45, x \approx 1.67$
However, we are given that $x \neq 2$, so the only valid solution is:
$x \approx 1.67$
Therefore, the solution to the equation is $x \approx 1.67$.
