%2F(6x)%2B(x%5E(2)-3x-10)%2F(6x)%3D(x-1)%2F(6).jpeg "(x-4)/(6x)+(x^(2)-3x-10)/(6x)=(x-1)/(6)")
Simplifying Algebraic Expressions
In this article, we will explore the simplification of a complex algebraic expression involving fractions.
The Given Expression
The expression we will be working with is:
$\frac{x-4}{6x} + \frac{x^2-3x-10}{6x} = \frac{x-1}{6}$
Step 1: Combine Like Terms
First, we will combine the two fractions on the left-hand side of the equation using a common denominator of $6x$:
$\frac{x-4}{6x} + \frac{x^2-3x-10}{6x} = \frac{x-4+x^2-3x-10}{6x}$
Step 2: Simplify the Numerator
Now, we will simplify the numerator of the combined fraction:
$x-4+x^2-3x-10 = x^2-2x-14$
So, the expression becomes:
$\frac{x^2-2x-14}{6x} = \frac{x-1}{6}$
Step 3: Cross-Multiply
To eliminate the fractions, we can cross-multiply:
$6x(x-1) = x^2-2x-14$
Step 4: Expand and Simplify
Expanding the left-hand side, we get:
$6x^2-6x = x^2-2x-14$
Subtracting $x^2$ from both sides yields:
$5x^2-6x = -14$
Final Result
Our final simplified expression is:
$5x^2-6x+14 = 0$
This is a quadratic equation that can be solved for $x$.
