Solve (5sqrt{2}-e)(3sqrt{2}+e)=fsqrt{2}-6

$$( 5 \sqrt { 2 } - e ) ( 3 \sqrt { 2 } + e ) = f \sqrt { 2 } - 6$$

Solve for f

$$f=-\frac{\sqrt{2}e^{2}}{2}+2e+18\sqrt{2}\approx 25.667556106$$

Steps for Solving Linear Equation

Apply the distributive property by multiplying each term of 5\sqrt{2}-e by each term of 3\sqrt{2}+e.

$$15\left(\sqrt{2}\right)^{2}+5\sqrt{2}e-3e\sqrt{2}-e^{2}=f\sqrt{2}-6$$

The square of \sqrt{2} is 2.

$$15\times 2+5\sqrt{2}e-3e\sqrt{2}-e^{2}=f\sqrt{2}-6$$

Multiply 15 and 2 to get 30.

$$30+5\sqrt{2}e-3e\sqrt{2}-e^{2}=f\sqrt{2}-6$$

Combine 5\sqrt{2}e and -3e\sqrt{2} to get 2\sqrt{2}e.

$$30+2\sqrt{2}e-e^{2}=f\sqrt{2}-6$$

Swap sides so that all variable terms are on the left hand side.

$$f\sqrt{2}-6=30+2\sqrt{2}e-e^{2}$$

Add 6 to both sides.

$$f\sqrt{2}=30+2\sqrt{2}e-e^{2}+6$$

Add 30 and 6 to get 36.

$$f\sqrt{2}=36+2\sqrt{2}e-e^{2}$$

The equation is in standard form.

$$\sqrt{2}f=2e\sqrt{2}-e^{2}+36$$

Divide both sides by \sqrt{2}.

$$\frac{\sqrt{2}f}{\sqrt{2}}=\frac{2e\sqrt{2}-e^{2}+36}{\sqrt{2}}$$

Dividing by \sqrt{2} undoes the multiplication by \sqrt{2}.

$$f=\frac{2e\sqrt{2}-e^{2}+36}{\sqrt{2}}$$

Divide 36+2e\sqrt{2}-e^{2} by \sqrt{2}.

$$f=\frac{\sqrt{2}\left(2e\sqrt{2}-e^{2}+36\right)}{2}$$