Solve A(w)=-w^2+100w

$$A ( w ) = - w ^ { 2 } + 100 w$$

Factor

$$w\left(100-w\right)$$

Solution Steps

Factor out w.

$$w\left(-w+100\right)$$

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.

$$-w^{2}+100w=0$$

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.

$$w=\frac{-100±\sqrt{100^{2}}}{2\left(-1\right)}$$

Take the square root of 100^{2}.

$$w=\frac{-100±100}{2\left(-1\right)}$$

Multiply 2 times -1.

$$w=\frac{-100±100}{-2}$$

Now solve the equation w=\frac{-100±100}{-2} when ± is plus. Add -100 to 100.

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

Divide 0 by -2.

$$w=0$$

Now solve the equation w=\frac{-100±100}{-2} when ± is minus. Subtract 100 from -100.

$$w=-\frac{200}{-2}$$

Divide -200 by -2.

$$w=100$$

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 100 for x_{2}.

$$-w^{2}+100w=-w\left(w-100\right)$$

Evaluate

$$w\left(100-w\right)$$