Solve f(x)=6x-x^2

$$f ( x ) = 6 x - x ^ { 2 }$$

Factor

$$x\left(6-x\right)$$

Solution Steps

Factor out x.

$$x\left(6-x\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-x^{2}+6x=0$$

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.

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

Take the square root of 6^{2}.

$$x=\frac{-6±6}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-6±6}{-2}$$

Now solve the equation x=\frac{-6±6}{-2} when ± is plus. Add -6 to 6.

$$x=\frac{0}{-2}$$

Divide 0 by -2.

$$x=0$$

Now solve the equation x=\frac{-6±6}{-2} when ± is minus. Subtract 6 from -6.

$$x=-\frac{12}{-2}$$

Divide -12 by -2.

$$x=6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 0 for x_{1} and 6 for x_{2}.

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

Evaluate

$$x\left(6-x\right)$$