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## Solve fx=x^2-6x+8 | Microsoft Math Solver

fx-x^{2}=-6x+8

Subtract x^{2} from both sides.

fx-x^{2}+6x=8

Add 6x to both sides.

fx-x^{2}+6x-8=0

Subtract 8 from both sides.

\left(f+6\right)x-x^{2}-8=0

Combine all terms containing x.

-x^{2}+\left(f+6\right)x-8=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.

x=\frac{-\left(f+6\right)±\sqrt{\left(f+6\right)^{2}-4\left(-1\right)\left(-8\right)}}{2\left(-1\right)}

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, f+6 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

x=\frac{-\left(f+6\right)±\sqrt{\left(f+6\right)^{2}+4\left(-8\right)}}{2\left(-1\right)}

Multiply -4 times -1.

x=\frac{-\left(f+6\right)±\sqrt{\left(f+6\right)^{2}-32}}{2\left(-1\right)}

Multiply 4 times -8.

x=\frac{-\left(f+6\right)±\sqrt{f^{2}+12f+4}}{2\left(-1\right)}

Take the square root of \left(f+6\right)^{2}-32.

x=\frac{-\left(f+6\right)±\sqrt{f^{2}+12f+4}}{-2}

Multiply 2 times -1.

x=\frac{\sqrt{f^{2}+12f+4}-f-6}{-2}

Now solve the equation x=\frac{-\left(f+6\right)±\sqrt{f^{2}+12f+4}}{-2} when ± is plus. Add -\left(f+6\right) to \sqrt{12f+4+f^{2}}.

x=-\frac{\sqrt{f^{2}+12f+4}}{2}+\frac{f}{2}+3

Divide -f-6+\sqrt{12f+4+f^{2}} by -2.

x=\frac{-\sqrt{f^{2}+12f+4}-f-6}{-2}

Now solve the equation x=\frac{-\left(f+6\right)±\sqrt{f^{2}+12f+4}}{-2} when ± is minus. Subtract \sqrt{12f+4+f^{2}} from -\left(f+6\right).

x=\frac{\sqrt{f^{2}+12f+4}}{2}+\frac{f}{2}+3

Divide -f-6-\sqrt{12f+4+f^{2}} by -2.

x=-\frac{\sqrt{f^{2}+12f+4}}{2}+\frac{f}{2}+3 x=\frac{\sqrt{f^{2}+12f+4}}{2}+\frac{f}{2}+3

The equation is now solved.

fx-x^{2}=-6x+8

Subtract x^{2} from both sides.

fx-x^{2}+6x=8

Add 6x to both sides.

\left(f+6\right)x-x^{2}=8

Combine all terms containing x.

-x^{2}+\left(f+6\right)x=8

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

\frac{-x^{2}+\left(f+6\right)x}{-1}=\frac{8}{-1}

Divide both sides by -1.

x^{2}+\frac{f+6}{-1}x=\frac{8}{-1}

Dividing by -1 undoes the multiplication by -1.

x^{2}+\left(-\left(f+6\right)\right)x=\frac{8}{-1}

Divide f+6 by -1.

x^{2}+\left(-\left(f+6\right)\right)x=-8

Divide 8 by -1.

x^{2}+\left(-\left(f+6\right)\right)x+\left(-\frac{f}{2}-3\right)^{2}=-8+\left(-\frac{f}{2}-3\right)^{2}

Divide -\left(f+6\right), the coefficient of the x term, by 2 to get -\frac{f}{2}-3. Then add the square of -\frac{f}{2}-3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

x^{2}+\left(-\left(f+6\right)\right)x+\frac{\left(f+6\right)^{2}}{4}=-8+\frac{\left(f+6\right)^{2}}{4}

Square -\frac{f}{2}-3.

x^{2}+\left(-\left(f+6\right)\right)x+\frac{\left(f+6\right)^{2}}{4}=\frac{\left(f+6\right)^{2}}{4}-8

Add -8 to \frac{\left(f+6\right)^{2}}{4}.

\left(x-\frac{f}{2}-3\right)^{2}=\frac{\left(f+6\right)^{2}}{4}-8

Factor x^{2}+\left(-\left(f+6\right)\right)x+\frac{\left(f+6\right)^{2}}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

\sqrt{\left(x-\frac{f}{2}-3\right)^{2}}=\sqrt{\frac{\left(f+6\right)^{2}}{4}-8}

Take the square root of both sides of the equation.

x-\frac{f}{2}-3=\frac{\sqrt{f^{2}+12f+4}}{2} x-\frac{f}{2}-3=-\frac{\sqrt{f^{2}+12f+4}}{2}

Simplify.

x=\frac{\sqrt{f^{2}+12f+4}}{2}+\frac{f}{2}+3 x=-\frac{\sqrt{f^{2}+12f+4}}{2}+\frac{f}{2}+3

Subtract -\frac{f}{2}-3 from both sides of the equation.

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