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\left(9a-20\right)^{2}=\left(\sqrt{400-a^{2}}\right)^{2}
Square both sides of the equation.
81a^{2}-360a+400=\left(\sqrt{400-a^{2}}\right)^{2}
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(9a-20\right)^{2}.
81a^{2}-360a+400=400-a^{2}
Calculate \sqrt{400-a^{2}} to the power of 2 and get 400-a^{2}.
81a^{2}-360a+400-400=-a^{2}
Subtract 400 from both sides.
81a^{2}-360a=-a^{2}
Subtract 400 from 400 to get 0.
81a^{2}-360a+a^{2}=0
Add a^{2} to both sides.
82a^{2}-360a=0
Combine 81a^{2} and a^{2} to get 82a^{2}.
a\left(82a-360\right)=0
Factor out a.
a=0 a=\frac{180}{41}
To find equation solutions, solve a=0 and 82a-360=0.
9\times 0-20=\sqrt{400-0^{2}}
Substitute 0 for a in the equation 9a-20=\sqrt{400-a^{2}}.
-20=20
Simplify. The value a=0 does not satisfy the equation because the left and the right hand side have opposite signs.
9\times \frac{180}{41}-20=\sqrt{400-\left(\frac{180}{41}\right)^{2}}
Substitute \frac{180}{41} for a in the equation 9a-20=\sqrt{400-a^{2}}.
\frac{800}{41}=\frac{800}{41}
Simplify. The value a=\frac{180}{41} satisfies the equation.
a=\frac{180}{41}
Equation 9a-20=\sqrt{400-a^{2}} has a unique solution.