Solve (18w+9)(8w-9)=(18w-9)(800+9)

$$( 18 w + 9 ) ( 8 w - 9 ) = ( 18 w - 9 ) ( 800 + 9 )$$

Solve for w

$$w = \frac{\sqrt{162449} + 407}{8} \approx 101.256203092$$

$$w=\frac{407-\sqrt{162449}}{8}\approx 0.493796908$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 18w+9 by 8w-9 and combine like terms.

$$144w^{2}-90w-81=\left(18w-9\right)\left(800+9\right)$$

Add 800 and 9 to get 809.

$$144w^{2}-90w-81=\left(18w-9\right)\times 809$$

Use the distributive property to multiply 18w-9 by 809.

$$144w^{2}-90w-81=14562w-7281$$

Subtract 14562w from both sides.

$$144w^{2}-90w-81-14562w=-7281$$

Combine -90w and -14562w to get -14652w.

$$144w^{2}-14652w-81=-7281$$

Add 7281 to both sides.

$$144w^{2}-14652w-81+7281=0$$

Add -81 and 7281 to get 7200.

$$144w^{2}-14652w+7200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 144 for a, -14652 for b, and 7200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-\left(-14652\right)±\sqrt{\left(-14652\right)^{2}-4\times 144\times 7200}}{2\times 144}$$

Square -14652.

$$w=\frac{-\left(-14652\right)±\sqrt{214681104-4\times 144\times 7200}}{2\times 144}$$

Multiply -4 times 144.

$$w=\frac{-\left(-14652\right)±\sqrt{214681104-576\times 7200}}{2\times 144}$$

Multiply -576 times 7200.

$$w=\frac{-\left(-14652\right)±\sqrt{214681104-4147200}}{2\times 144}$$

Add 214681104 to -4147200.

$$w=\frac{-\left(-14652\right)±\sqrt{210533904}}{2\times 144}$$

Take the square root of 210533904.

$$w=\frac{-\left(-14652\right)±36\sqrt{162449}}{2\times 144}$$

The opposite of -14652 is 14652.

$$w=\frac{14652±36\sqrt{162449}}{2\times 144}$$

Multiply 2 times 144.

$$w=\frac{14652±36\sqrt{162449}}{288}$$

Now solve the equation w=\frac{14652±36\sqrt{162449}}{288} when ± is plus. Add 14652 to 36\sqrt{162449}.

$$w=\frac{36\sqrt{162449}+14652}{288}$$

Divide 14652+36\sqrt{162449} by 288.

$$w=\frac{\sqrt{162449}+407}{8}$$

Now solve the equation w=\frac{14652±36\sqrt{162449}}{288} when ± is minus. Subtract 36\sqrt{162449} from 14652.

$$w=\frac{14652-36\sqrt{162449}}{288}$$

Divide 14652-36\sqrt{162449} by 288.

$$w=\frac{407-\sqrt{162449}}{8}$$

The equation is now solved.

$$w=\frac{\sqrt{162449}+407}{8} w=\frac{407-\sqrt{162449}}{8}$$

Steps for Completing the Square

Use the distributive property to multiply 18w+9 by 8w-9 and combine like terms.

$$144w^{2}-90w-81=\left(18w-9\right)\left(800+9\right)$$

Add 800 and 9 to get 809.

$$144w^{2}-90w-81=\left(18w-9\right)\times 809$$

Use the distributive property to multiply 18w-9 by 809.

$$144w^{2}-90w-81=14562w-7281$$

Subtract 14562w from both sides.

$$144w^{2}-90w-81-14562w=-7281$$

Combine -90w and -14562w to get -14652w.

$$144w^{2}-14652w-81=-7281$$

Add 81 to both sides.

$$144w^{2}-14652w=-7281+81$$

Add -7281 and 81 to get -7200.

$$144w^{2}-14652w=-7200$$

Divide both sides by 144.

$$\frac{144w^{2}-14652w}{144}=-\frac{7200}{144}$$

Dividing by 144 undoes the multiplication by 144.

$$w^{2}+\left(-\frac{14652}{144}\right)w=-\frac{7200}{144}$$

Reduce the fraction \frac{-14652}{144} to lowest terms by extracting and canceling out 36.

$$w^{2}-\frac{407}{4}w=-\frac{7200}{144}$$

Divide -7200 by 144.

$$w^{2}-\frac{407}{4}w=-50$$

Divide -\frac{407}{4}, the coefficient of the x term, by 2 to get -\frac{407}{8}. Then add the square of -\frac{407}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}-\frac{407}{4}w+\left(-\frac{407}{8}\right)^{2}=-50+\left(-\frac{407}{8}\right)^{2}$$

Square -\frac{407}{8} by squaring both the numerator and the denominator of the fraction.

$$w^{2}-\frac{407}{4}w+\frac{165649}{64}=-50+\frac{165649}{64}$$

Add -50 to \frac{165649}{64}.

$$w^{2}-\frac{407}{4}w+\frac{165649}{64}=\frac{162449}{64}$$

Factor w^{2}-\frac{407}{4}w+\frac{165649}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(w-\frac{407}{8}\right)^{2}=\frac{162449}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w-\frac{407}{8}\right)^{2}}=\sqrt{\frac{162449}{64}}$$

Simplify.

$$w-\frac{407}{8}=\frac{\sqrt{162449}}{8} w-\frac{407}{8}=-\frac{\sqrt{162449}}{8}$$

Add \frac{407}{8} to both sides of the equation.

$$w=\frac{\sqrt{162449}+407}{8} w=\frac{407-\sqrt{162449}}{8}$$