Solve lambda^3-10lambda^2+left(34-3aright)lambda-36+6a=0

$${ \lambda }^{ 3 } -10 { \lambda }^{ 2 } + \left( 34-3a \right) \lambda -36+6a = 0$$

Solve for a

$$\left\{\begin{matrix}\\a=\frac{\lambda ^{2}-8\lambda +18}{3}\text{, }&\text{unconditionally}\\a\in \mathrm{R}\text{, }&\lambda =2\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 34-3a by \lambda .

$$\lambda ^{3}-10\lambda ^{2}+34\lambda -3a\lambda -36+6a=0$$

Subtract \lambda ^{3} from both sides. Anything subtracted from zero gives its negation.

$$-10\lambda ^{2}+34\lambda -3a\lambda -36+6a=-\lambda ^{3}$$

Add 10\lambda ^{2} to both sides.

$$34\lambda -3a\lambda -36+6a=-\lambda ^{3}+10\lambda ^{2}$$

Subtract 34\lambda from both sides.

$$-3a\lambda -36+6a=-\lambda ^{3}+10\lambda ^{2}-34\lambda$$

Add 36 to both sides.

$$-3a\lambda +6a=-\lambda ^{3}+10\lambda ^{2}-34\lambda +36$$

Combine all terms containing a.

$$\left(-3\lambda +6\right)a=-\lambda ^{3}+10\lambda ^{2}-34\lambda +36$$

The equation is in standard form.

$$\left(6-3\lambda \right)a=36-34\lambda +10\lambda ^{2}-\lambda ^{3}$$

Divide both sides by -3\lambda +6.

$$\frac{\left(6-3\lambda \right)a}{6-3\lambda }=\frac{\left(\lambda -2\right)\left(-\lambda ^{2}+8\lambda -18\right)}{6-3\lambda }$$

Dividing by -3\lambda +6 undoes the multiplication by -3\lambda +6.

$$a=\frac{\left(\lambda -2\right)\left(-\lambda ^{2}+8\lambda -18\right)}{6-3\lambda }$$

Divide \left(-18+8\lambda -\lambda ^{2}\right)\left(-2+\lambda \right) by -3\lambda +6.

$$a=\frac{\lambda ^{2}}{3}-\frac{8\lambda }{3}+6$$

Solve for λ

$$\left\{\begin{matrix}\\\lambda =2\text{, }&\text{unconditionally}\\\lambda =\sqrt{3a-2}+4\text{; }\lambda =-\sqrt{3a-2}+4\text{, }&a\geq \frac{2}{3}\end{matrix}\right.$$