Solve for r. (1)/(r-2)+(1)/(r^(2)-7r+10)=(6)/(r-2)

Factor Quadratic Expression: Factor the quadratic expression in the denominator. The quadratic expression r^2 - 7r + 10 can be factored into (r - 5)(r - 2). r^2 - 7r + 10 = (r - 5)(r - 2)Write with Factored Denominator: Write the given expression with the factored denominator. (1)/(r-2) + (1)/((r-5)(r-2)) = (6)/(r-2)Find Common Denominator: Find a common denominator for the two fractions on the left side of the equation. The common denominator is (r - 5)(r - 2).Rewrite with Common Denominator: Rewrite each fraction with the common denominator. (1)/(r-2) * ((r - 5)/(r - 5)) + (1)/((r-5)(r-2)) = (6)/(r-2) This becomes: ((r - 5) + 1)/((r-5)(r-2)) = (6)/(r-2)Combine Numerators: Combine the numerators of the fractions on the left side. ((r - 5) + 1)/((r-5)(r-2)) = (r - 5 + 1)/((r-5)(r-2)) This simplifies to: (r - 4)/((r-5)(r-2)) = (6)/(r-2)Equate Numerators: Since the denominators on both sides of the equation are the same, we can equate the numerators. (r - 4) = 6Solve for r: Solve for r. r - 4 = 6 r = 6 + 4 r = 10

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Solve for $r$ . $\newline$ $\left(\frac{1}{r-2}\right)+\left(\frac{1}{r^{2}-7r+10}\right)=\left(\frac{6}{r-2}\right)$

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Algebra 2

Add, subtract, multiply, and divide polynomials

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Q. Solve for

r

\newline

\left(\frac{1}{r-2}\right)+\left(\frac{1}{r^{2}-7r+10}\right)=\left(\frac{6}{r-2}\right)

Factor Quadratic Expression: Factor the quadratic expression in the denominator. $\newline$ The quadratic expression $r^2 - 7r + 10$ can be factored into $(r - 5)(r - 2)$ . $\newline$ $r^2 - 7r + 10 = (r - 5)(r - 2)$
Write with Factored Denominator: Write the given expression with the factored denominator. $\newline$ $(1)/(r-2) + (1)/((r-5)(r-2)) = (6)/(r-2)$
Find Common Denominator: Find a common denominator for the two fractions on the left side of the equation. $\newline$ The common denominator is $(r - 5)(r - 2)$ .
Rewrite with Common Denominator: Rewrite each fraction with the common denominator. $\newline$ $(1)/(r-2) \times ((r - 5)/(r - 5)) + (1)/((r-5)(r-2)) = (6)/(r-2)$ $\newline$ This becomes: $\newline$ $((r - 5) + 1)/((r-5)(r-2)) = (6)/(r-2)$
Combine Numerators: Combine the numerators of the fractions on the left side. $\newline$ $\frac{(r - 5) + 1}{(r-5)(r-2)} = \frac{r - 5 + 1}{(r-5)(r-2)}$ $\newline$ This simplifies to: $\newline$ $\frac{r - 4}{(r-5)(r-2)} = \frac{6}{r-2}$
Equate Numerators: Since the denominators on both sides of the equation are the same, we can equate the numerators. $\newline$ $(r - 4) = 6$
Solve for r: Solve for r. $\newline$ $r - 4 = 6$ $\newline$ $r = 6 + 4$ $\newline$ $r = 10$