Solve sqrt{2}xleft(4-aright)=left(3a+6right)left(a+4-1/2a^2right)

$$\sqrt{ 2 } x \left( 4-a \right) = \left( 3a+6 \right) \left( a+4- \frac{ 1 }{ 2 } { a }^{ 2 } \right)$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\frac{3\sqrt{2}\left(a+2\right)^{2}}{4}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&a=4\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \sqrt{2}x by 4-a.

$$4\sqrt{2}x-\sqrt{2}xa=\left(3a+6\right)\left(a+4-\frac{1}{2}a^{2}\right)$$

Use the distributive property to multiply 3a+6 by a+4-\frac{1}{2}a^{2} and combine like terms.

$$4\sqrt{2}x-\sqrt{2}xa=18a-\frac{3}{2}a^{3}+24$$

Combine all terms containing x.

$$\left(4\sqrt{2}-\sqrt{2}a\right)x=18a-\frac{3}{2}a^{3}+24$$

The equation is in standard form.

$$\left(-\sqrt{2}a+4\sqrt{2}\right)x=-\frac{3a^{3}}{2}+18a+24$$

Divide both sides by 4\sqrt{2}-\sqrt{2}a.

$$\frac{\left(-\sqrt{2}a+4\sqrt{2}\right)x}{-\sqrt{2}a+4\sqrt{2}}=\frac{-\frac{3a^{3}}{2}+18a+24}{-\sqrt{2}a+4\sqrt{2}}$$

Dividing by 4\sqrt{2}-\sqrt{2}a undoes the multiplication by 4\sqrt{2}-\sqrt{2}a.

$$x=\frac{-\frac{3a^{3}}{2}+18a+24}{-\sqrt{2}a+4\sqrt{2}}$$

Divide 18a-\frac{3a^{3}}{2}+24 by 4\sqrt{2}-\sqrt{2}a.

$$x=\frac{3\sqrt{2}\left(a+2\right)^{2}}{4}$$

Solve for x

$$\left\{\begin{matrix}\\x=\frac{3\sqrt{2}\left(a+2\right)^{2}}{4}\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&a=4\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \sqrt{2}x by 4-a.

$$4\sqrt{2}x-\sqrt{2}xa=\left(3a+6\right)\left(a+4-\frac{1}{2}a^{2}\right)$$

Use the distributive property to multiply 3a+6 by a+4-\frac{1}{2}a^{2} and combine like terms.

$$4\sqrt{2}x-\sqrt{2}xa=18a-\frac{3}{2}a^{3}+24$$

Combine all terms containing x.

$$\left(4\sqrt{2}-\sqrt{2}a\right)x=18a-\frac{3}{2}a^{3}+24$$

The equation is in standard form.

$$\left(-\sqrt{2}a+4\sqrt{2}\right)x=-\frac{3a^{3}}{2}+18a+24$$

Divide both sides by 4\sqrt{2}-\sqrt{2}a.

$$\frac{\left(-\sqrt{2}a+4\sqrt{2}\right)x}{-\sqrt{2}a+4\sqrt{2}}=\frac{-\frac{3a^{3}}{2}+18a+24}{-\sqrt{2}a+4\sqrt{2}}$$

Dividing by 4\sqrt{2}-\sqrt{2}a undoes the multiplication by 4\sqrt{2}-\sqrt{2}a.

$$x=\frac{-\frac{3a^{3}}{2}+18a+24}{-\sqrt{2}a+4\sqrt{2}}$$

Divide 18a-\frac{3a^{3}}{2}+24 by 4\sqrt{2}-\sqrt{2}a.

$$x=\frac{3\sqrt{2}\left(a+2\right)^{2}}{4}$$

Solve for a (complex solution)

$$a=\frac{2^{\frac{3}{4}}\sqrt{3x}}{3}-2$$

$$a=4$$

$$a=-\frac{2^{\frac{3}{4}}\sqrt{3x}}{3}-2$$