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${:[(x-6)/(x^(2)-5x+6)=(A)/(x-2)+(B)/(x-3)],[=>5A-B=?]:}$

$55$ . $\newline$ $\begin{array}{l} \frac{x-6}{x^{2}-5 x+6}=\frac{A}{x-2}+\frac{B}{x-3} \\ \Rightarrow 5 A-B=? \end{array}$

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Evaluate integers raised to rational exponents

Full solution

Q.

55

.

\newline

\begin{array}{l} \frac{x-6}{x^{2}-5 x+6}=\frac{A}{x-2}+\frac{B}{x-3} \\ \Rightarrow 5 A-B=? \end{array}

Factor Denominator: First, factor the denominator $x^2 - 5x + 6$ .$x^2 - 5x + 6 = (x - 2)(x - 3)
Set Up Equation: Now, set up the equation for partial fraction decomposition. $\newline$ $(x - 6) / [(x - 2)(x - 3)] = A / (x - 2) + B / (x - 3)$
Clear Fractions: Multiply both sides by the common denominator $(x - 2)(x - 3)$ to clear the fractions. $\newline$ $(x - 6) = A(x - 3) + B(x - 2)$
Expand Equation: Expand the right side of the equation. $\newline$ $x - 6 = Ax - 3A + Bx - 2B$
Combine Like Terms: Combine like terms. $x - 6 = (A + B)x - (3A + 2B)$
Set Up System: Set up a system of equations by matching coefficients from both sides of the equation. $\newline$ $1$ . For the $x$ terms: $A + B = 1$ $\newline$ $2$ . For the constant terms: $-3A - 2B = -6$
Solve System: Solve the system of equations. Start with the first equation. $\newline$ $A + B = 1$ $\newline$ Let's solve for $A$ : $A = 1 - B$
Substitute A: Substitute $A = 1 - B$ into the second equation. $\newline$ $-3(1 - B) - 2B = -6$
Distribute and Combine: Distribute and combine like terms. $-3 + 3B - 2B = -6$
Simplify Equation: Simplify the equation. $B = -6 + 3$

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