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Solve
(a)/(x+a)+(b)/(x-b)=(a+b)/(x+c) for
x

Solve $\frac{a}{x+a}+\frac{b}{x-b}=\frac{a+b}{x+c}$ for $x$

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Write a quadratic function from its x-intercepts and another point

Full solution

Q. Solve

\frac{a}{x+a}+\frac{b}{x-b}=\frac{a+b}{x+c}

for

x

Rewrite with common denominator: Now, rewrite each fraction with the common denominator: $\frac{a(x-b)}{(x+a)(x-b)} + \frac{b(x+a)}{(x+a)(x-b)} = \frac{a+b}{x+c}$ .
Combine fractions: Combine the fractions on the left side: $\frac{(ax - ab) + (bx + ab)}{(x+a)(x-b)} = \frac{a+b}{x+c}$ .
Simplify numerator: Simplify the numerator on the left side: $(ax + bx)/(x+a)(x-b) = (a+b)/(x+c)$ .
Combine like terms: Combine like terms in the numerator: $(a+b)x/(x+a)(x-b) = (a+b)/(x+c)$ .
Cancel out common factor: Since $(a+b)$ appears in both numerators, we can cancel it out, assuming $a+b \neq 0$ : $\frac{x}{(x+a)(x-b)} = \frac{1}{(x+c)}.$
Cross-multiply: Cross-multiply to get rid of the fractions: $x(x+c) = (x+a)(x-b)$ .
Expand both sides: Expand both sides: $x^2 + cx = x^2 + ax - bx - ab$ .
Subtract $x^2$ : Subtract $x^2$ from both sides to simplify: $cx = ax - bx - ab$ .
Combine like terms: Combine like terms: $cx = (a - b)x - ab$ .
Subtract $(a - b)x$ : Subtract $(a - b)x$ from both sides: $\newline$ $cx - (a - b)x = -ab$ .
Factor out $x$ : Factor out $x$ on the left side: $\newline$ $x(c - a + b) = -ab$ .
Divide to solve for $x$ : Divide both sides by $(c - a + b)$ to solve for $x$ : $x = \frac{-ab}{(c - a + b)}.$

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