Each value of a lies on a circle of some radius and centered on the origin of the complex plane. For √a+√(-a) to have no imaginary component, a must be purely imaginary, so set a=±ri where r>0 is real and i=√(-1). Thus, ±a=re^{±iπ/2} and √(±a)=√re^{±iπ/4}=√r[cos(π/4)±i*sin(π/4)]=√r(√2/2)(1±i). It follows that √a+√(-a)=√(2r)=10 or 2r=100 and r=50. Thus, a=±50i.
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Each value of a lies on a circle of some radius and centered on the origin of the complex plane. For √a+√(-a) to have no imaginary component, a must be purely imaginary, so set a=±ri where r>0 is real and i=√(-1). Thus, ±a=re^{±iπ/2} and √(±a)=√re^{±iπ/4}=√r[cos(π/4)±i*sin(π/4)]=√r(√2/2)(1±i). It follows that √a+√(-a)=√(2r)=10 or 2r=100 and r=50. Thus, a=±50i.
Yeah
Hardest problem? It's basic algebra.