Solve R(p)=-9p^2+18.000p

$$R ( p ) = - 9 p ^ { 2 } + 18.000 p$$

Factor

$$9p\left(2-p\right)$$

Solution Steps

Factor out 9.

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

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

$$p\left(-p+2\right)$$

Rewrite the complete factored expression.

$$9p\left(-p+2\right)$$

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.

$$-9p^{2}+18p=0$$

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.

$$p=\frac{-18±\sqrt{18^{2}}}{2\left(-9\right)}$$

Take the square root of 18^{2}.

$$p=\frac{-18±18}{2\left(-9\right)}$$

Multiply 2 times -9.

$$p=\frac{-18±18}{-18}$$

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

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

Divide 0 by -18.

$$p=0$$

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

$$p=-\frac{36}{-18}$$

Divide -36 by -18.

$$p=2$$

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 2 for x_{2}.

$$-9p^{2}+18p=-9p\left(p-2\right)$$

Evaluate

$$9p\left(2-p\right)$$