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solve $y=\frac{\sqrt{ax+8}}{3+x}$ for $x$

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Full solution

Q. solve

y=\frac{\sqrt{ax+8}}{3+x}

for

x

Square both sides: Square both sides to get rid of the square root. $y^2 = \frac{ax + 8}{3 + x}$
Multiply by $(3 + x)$ : Multiply both sides by $(3 + x)$ to clear the fraction. $\newline$ $y^2(3 + x) = ax + 8$
Distribute $y^2$ : Distribute $y^2$ on the left side. $\newline$ $3y^2 + xy^2 = ax + 8$
Subtract $xy^2$ : Subtract $xy^2$ from both sides to get terms with $x$ on one side. $\newline$ $3y^2 = ax + 8 - xy^2$
Factor out x: Factor out x on the right side. $\newline$ $3y^2 = x(a - y^2) + 8$
Subtract $8$ : Subtract $8$ from both sides. $\newline$ $3y^2 - 8 = x(a - y^2)$
Divide by $(a - y^2)$ : Divide both sides by $(a - y^2)$ to solve for $x$ . $x = \frac{3y^2 - 8}{a - y^2}$

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