1. **Problem statement:** Convert the given expression into partial fractions and find the value of $x$. 2. **General formula and rules:** Partial fraction decomposition is used to express a rational function as a sum of simpler fractions. Typically, for a rational function $\frac{P(x)}{Q(x)}$, where degree of $P(x)$ is less than degree of $Q(x)$, we write: $$\frac{P(x)}{Q(x)} = \sum \frac{A}{(linear\ factors)} + \sum \frac{Bx + C}{(quadratic\ factors)}$$ 3. **Step-by-step solution:** - Since the exact expression is not provided, let's assume the expression is $\frac{N(x)}{D(x)}$. - Factorize the denominator $D(x)$ into linear or irreducible quadratic factors. - Set up the partial fraction form with unknown coefficients. - Multiply both sides by $D(x)$ to clear denominators. - Equate coefficients of powers of $x$ or substitute convenient values of $x$ to solve for unknowns. - Once coefficients are found, if the problem asks to find $x$ for a certain condition, substitute back and solve. 4. **Example:** Suppose the expression is $\frac{3x+5}{(x-1)(x+2)}$. - Partial fractions: $\frac{3x+5}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$. - Multiply both sides by $(x-1)(x+2)$: $$3x+5 = A(x+2) + B(x-1)$$ - Expand: $$3x+5 = Ax + 2A + Bx - B = (A+B)x + (2A - B)$$ - Equate coefficients: $$3 = A + B$$ $$5 = 2A - B$$ - Solve the system: Add equations: $3 + 5 = (A+B) + (2A - B) = 3A \Rightarrow 8 = 3A \Rightarrow A = \frac{8}{3}$ - Substitute $A$ into $3 = A + B$: $$3 = \frac{8}{3} + B \Rightarrow B = 3 - \frac{8}{3} = \frac{9}{3} - \frac{8}{3} = \frac{1}{3}$$ 5. **Final answer:** $$\frac{3x+5}{(x-1)(x+2)} = \frac{8/3}{x-1} + \frac{1/3}{x+2}$$ If the problem asks to find $x$ for a certain value, substitute and solve accordingly. Since the original expression and condition to find $x$ are not provided, this is the general method and example for partial fraction decomposition and solving for $x$.