Cubic Equation and Inequality

Algebra Level 5

(λ+3)2+(ρ+3)2+18+3λ+3ρ+1λ2+ρ2+17 \large \frac{\sqrt{(\lambda +3)^2 + (\rho +3)^2 +18} + 3\lambda +3\rho +1}{\sqrt{\lambda^2 + \rho^2} +1} \geq 7

Let λ,ϵ,ρ\lambda, \epsilon, \rho be the roots of the equation (x+a)3=0(x+a)^3 = 0 for real number aa. If the inequality above is fulfilled, find the numeric value of the expression below.

(λ+3)2+(ρ+3)2+18+3λ+3ρ+1λ2+ρ2+1 \large \frac{\sqrt{(\lambda +3)^2 + (\rho +3)^2 +18} + 3\lambda +3\rho +1}{\sqrt{\lambda^2 + \rho^2} +1}

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