Solve R(p)=-9p^2+18,000p

$$R ( p ) = - 9 p ^ { 2 } + 18,000 p$$

Factor

$$9p\left(2000-p\right)$$

Solution Steps

Factor out 9.

$$9\left(-p^{2}+2000p\right)$$

Consider -p^{2}+2000p. Factor out p.

$$p\left(-p+2000\right)$$

Rewrite the complete factored expression.

$$9p\left(-p+2000\right)$$

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.

$$-9p^{2}+18000p=0$$

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.

$$p=\frac{-18000±\sqrt{18000^{2}}}{2\left(-9\right)}$$

Take the square root of 18000^{2}.

$$p=\frac{-18000±18000}{2\left(-9\right)}$$

Multiply 2 times -9.

$$p=\frac{-18000±18000}{-18}$$

Now solve the equation p=\frac{-18000±18000}{-18} when ± is plus. Add -18000 to 18000.

$$p=\frac{0}{-18}$$

Divide 0 by -18.

$$p=0$$

Now solve the equation p=\frac{-18000±18000}{-18} when ± is minus. Subtract 18000 from -18000.

$$p=-\frac{36000}{-18}$$

Divide -36000 by -18.

$$p=2000$$

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 2000 for x_{2}.

$$-9p^{2}+18000p=-9p\left(p-2000\right)$$

Evaluate

$$9p\left(2000-p\right)$$