Solve 8/f-f/4=4/2

$$\frac { 8 } { f } - \frac { f } { 4 } = \frac { 4 } { 2 }$$

Solve for f

$$f=4\sqrt{3}-4\approx 2.92820323$$

$$f=-4\sqrt{3}-4\approx -10.92820323$$

Steps Using the Quadratic Formula

Variable f cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4f, the least common multiple of f,4,2.

$$4\times 8-ff=2f\times 4$$

Multiply f and f to get f^{2}.

$$4\times 8-f^{2}=2f\times 4$$

Multiply 4 and 8 to get 32.

$$32-f^{2}=2f\times 4$$

Multiply 2 and 4 to get 8.

$$32-f^{2}=8f$$

Subtract 8f from both sides.

$$32-f^{2}-8f=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-f^{2}-8f+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -8 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$f=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-1\right)\times 32}}{2\left(-1\right)}$$

Square -8.

$$f=\frac{-\left(-8\right)±\sqrt{64-4\left(-1\right)\times 32}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$f=\frac{-\left(-8\right)±\sqrt{64+4\times 32}}{2\left(-1\right)}$$

Multiply 4 times 32.

$$f=\frac{-\left(-8\right)±\sqrt{64+128}}{2\left(-1\right)}$$

Add 64 to 128.

$$f=\frac{-\left(-8\right)±\sqrt{192}}{2\left(-1\right)}$$

Take the square root of 192.

$$f=\frac{-\left(-8\right)±8\sqrt{3}}{2\left(-1\right)}$$

The opposite of -8 is 8.

$$f=\frac{8±8\sqrt{3}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$f=\frac{8±8\sqrt{3}}{-2}$$

Now solve the equation f=\frac{8±8\sqrt{3}}{-2} when ± is plus. Add 8 to 8\sqrt{3}.

$$f=\frac{8\sqrt{3}+8}{-2}$$

Divide 8+8\sqrt{3} by -2.

$$f=-4\sqrt{3}-4$$

Now solve the equation f=\frac{8±8\sqrt{3}}{-2} when ± is minus. Subtract 8\sqrt{3} from 8.

$$f=\frac{8-8\sqrt{3}}{-2}$$

Divide 8-8\sqrt{3} by -2.

$$f=4\sqrt{3}-4$$

The equation is now solved.

$$f=-4\sqrt{3}-4 f=4\sqrt{3}-4$$

Steps for Completing the Square

Variable f cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4f, the least common multiple of f,4,2.

$$4\times 8-ff=2f\times 4$$

Multiply f and f to get f^{2}.

$$4\times 8-f^{2}=2f\times 4$$

Multiply 4 and 8 to get 32.

$$32-f^{2}=2f\times 4$$

Multiply 2 and 4 to get 8.

$$32-f^{2}=8f$$

Subtract 8f from both sides.

$$32-f^{2}-8f=0$$

Subtract 32 from both sides. Anything subtracted from zero gives its negation.

$$-f^{2}-8f=-32$$

Divide both sides by -1.

$$\frac{-f^{2}-8f}{-1}=-\frac{32}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$f^{2}+\left(-\frac{8}{-1}\right)f=-\frac{32}{-1}$$

Divide -8 by -1.

$$f^{2}+8f=-\frac{32}{-1}$$

Divide -32 by -1.

$$f^{2}+8f=32$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$f^{2}+8f+4^{2}=32+4^{2}$$

Square 4.

$$f^{2}+8f+16=32+16$$

Add 32 to 16.

$$f^{2}+8f+16=48$$

Factor f^{2}+8f+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(f+4\right)^{2}=48$$

Take the square root of both sides of the equation.

$$\sqrt{\left(f+4\right)^{2}}=\sqrt{48}$$

Simplify.

$$f+4=4\sqrt{3} f+4=-4\sqrt{3}$$

Subtract 4 from both sides of the equation.

$$f=4\sqrt{3}-4 f=-4\sqrt{3}-4$$