Solve for a (complex solution)
a=\frac{\sqrt{x+2\sqrt{x-3}-2}+14\sqrt[4]{x-3}+32}{2\left(\sqrt[4]{x-3}+3\right)}
Solve for a
a=\frac{\sqrt{x+2\sqrt{x-3}-2}+14\sqrt[4]{x-3}+32}{2\left(\sqrt[4]{x-3}+3\right)}
x\geq 3
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\sqrt{x-2+2\sqrt{x-3}}+14\sqrt[4]{x-3}-2a\sqrt[4]{x-3}+32=6a
Use the distributive property to multiply 14-2a by \sqrt[4]{x-3}.
\sqrt{x-2+2\sqrt{x-3}}+14\sqrt[4]{x-3}-2a\sqrt[4]{x-3}+32-6a=0
Subtract 6a from both sides.
14\sqrt[4]{x-3}-2a\sqrt[4]{x-3}+32-6a=-\sqrt{x-2+2\sqrt{x-3}}
Subtract \sqrt{x-2+2\sqrt{x-3}} from both sides. Anything subtracted from zero gives its negation.
-2a\sqrt[4]{x-3}+32-6a=-\sqrt{x-2+2\sqrt{x-3}}-14\sqrt[4]{x-3}
Subtract 14\sqrt[4]{x-3} from both sides.
-2a\sqrt[4]{x-3}-6a=-\sqrt{x-2+2\sqrt{x-3}}-14\sqrt[4]{x-3}-32
Subtract 32 from both sides.
\left(-2\sqrt[4]{x-3}-6\right)a=-\sqrt{x-2+2\sqrt{x-3}}-14\sqrt[4]{x-3}-32
Combine all terms containing a.
\left(-2\sqrt[4]{x-3}-6\right)a=-\sqrt{x+2\sqrt{x-3}-2}-14\sqrt[4]{x-3}-32
The equation is in standard form.
\frac{\left(-2\sqrt[4]{x-3}-6\right)a}{-2\sqrt[4]{x-3}-6}=\frac{-\sqrt{x+2\sqrt{x-3}-2}-14\sqrt[4]{x-3}-32}{-2\sqrt[4]{x-3}-6}
Divide both sides by -2\sqrt[4]{x-3}-6.
a=\frac{-\sqrt{x+2\sqrt{x-3}-2}-14\sqrt[4]{x-3}-32}{-2\sqrt[4]{x-3}-6}
Dividing by -2\sqrt[4]{x-3}-6 undoes the multiplication by -2\sqrt[4]{x-3}-6.
a=\frac{\sqrt{x+2\sqrt{x-3}-2}+14\sqrt[4]{x-3}+32}{2\left(\sqrt[4]{x-3}+3\right)}
Divide -\sqrt{x-2+2\sqrt{x-3}}-14\sqrt[4]{x-3}-32 by -2\sqrt[4]{x-3}-6.
\sqrt{x-2+2\sqrt{x-3}}+14\sqrt[4]{x-3}-2a\sqrt[4]{x-3}+32=6a
Use the distributive property to multiply 14-2a by \sqrt[4]{x-3}.
\sqrt{x-2+2\sqrt{x-3}}+14\sqrt[4]{x-3}-2a\sqrt[4]{x-3}+32-6a=0
Subtract 6a from both sides.
14\sqrt[4]{x-3}-2a\sqrt[4]{x-3}+32-6a=-\sqrt{x-2+2\sqrt{x-3}}
Subtract \sqrt{x-2+2\sqrt{x-3}} from both sides. Anything subtracted from zero gives its negation.
-2a\sqrt[4]{x-3}+32-6a=-\sqrt{x-2+2\sqrt{x-3}}-14\sqrt[4]{x-3}
Subtract 14\sqrt[4]{x-3} from both sides.
-2a\sqrt[4]{x-3}-6a=-\sqrt{x-2+2\sqrt{x-3}}-14\sqrt[4]{x-3}-32
Subtract 32 from both sides.
\left(-2\sqrt[4]{x-3}-6\right)a=-\sqrt{x-2+2\sqrt{x-3}}-14\sqrt[4]{x-3}-32
Combine all terms containing a.
\left(-2\sqrt[4]{x-3}-6\right)a=-\sqrt{x+2\sqrt{x-3}-2}-14\sqrt[4]{x-3}-32
The equation is in standard form.
\frac{\left(-2\sqrt[4]{x-3}-6\right)a}{-2\sqrt[4]{x-3}-6}=\frac{-\sqrt{x+2\sqrt{x-3}-2}-14\sqrt[4]{x-3}-32}{-2\sqrt[4]{x-3}-6}
Divide both sides by -2\sqrt[4]{x-3}-6.
a=\frac{-\sqrt{x+2\sqrt{x-3}-2}-14\sqrt[4]{x-3}-32}{-2\sqrt[4]{x-3}-6}
Dividing by -2\sqrt[4]{x-3}-6 undoes the multiplication by -2\sqrt[4]{x-3}-6.
a=\frac{\sqrt{x+2\sqrt{x-3}-2}+14\sqrt[4]{x-3}+32}{2\left(\sqrt[4]{x-3}+3\right)}
Divide -\sqrt{x-2+2\sqrt{x-3}}-14\sqrt[4]{x-3}-32 by -2\sqrt[4]{x-3}-6.
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