Solve f(x)=72x-6x^2

$$f ( x ) = 72 x - 6 x ^ { 2 }$$

Factor

$$6x\left(12-x\right)$$

Solution Steps

Factor out 6.

$$6\left(12x-x^{2}\right)$$

Consider 12x-x^{2}. Factor out x.

$$x\left(12-x\right)$$

Rewrite the complete factored expression.

$$6x\left(-x+12\right)$$

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.

$$-6x^{2}+72x=0$$

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.

$$x=\frac{-72±\sqrt{72^{2}}}{2\left(-6\right)}$$

Take the square root of 72^{2}.

$$x=\frac{-72±72}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-72±72}{-12}$$

Now solve the equation x=\frac{-72±72}{-12} when ± is plus. Add -72 to 72.

$$x=\frac{0}{-12}$$

Divide 0 by -12.

$$x=0$$

Now solve the equation x=\frac{-72±72}{-12} when ± is minus. Subtract 72 from -72.

$$x=-\frac{144}{-12}$$

Divide -144 by -12.

$$x=12$$

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 12 for x_{2}.

$$-6x^{2}+72x=-6x\left(x-12\right)$$

Evaluate

$$6x\left(12-x\right)$$