Solve 44=w(2w-3)

$$44 = w ( 2 w - 3 )$$

Solve for w

$$w=-4$$

$$w = \frac{11}{2} = 5\frac{1}{2} = 5.5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply w by 2w-3.

$$44=2w^{2}-3w$$

Swap sides so that all variable terms are on the left hand side.

$$2w^{2}-3w=44$$

Subtract 44 from both sides.

$$2w^{2}-3w-44=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -3 for b, and -44 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2\left(-44\right)}}{2\times 2}$$

Square -3.

$$w=\frac{-\left(-3\right)±\sqrt{9-4\times 2\left(-44\right)}}{2\times 2}$$

Multiply -4 times 2.

$$w=\frac{-\left(-3\right)±\sqrt{9-8\left(-44\right)}}{2\times 2}$$

Multiply -8 times -44.

$$w=\frac{-\left(-3\right)±\sqrt{9+352}}{2\times 2}$$

Add 9 to 352.

$$w=\frac{-\left(-3\right)±\sqrt{361}}{2\times 2}$$

Take the square root of 361.

$$w=\frac{-\left(-3\right)±19}{2\times 2}$$

The opposite of -3 is 3.

$$w=\frac{3±19}{2\times 2}$$

Multiply 2 times 2.

$$w=\frac{3±19}{4}$$

Now solve the equation w=\frac{3±19}{4} when ± is plus. Add 3 to 19.

$$w=\frac{22}{4}$$

Reduce the fraction \frac{22}{4} to lowest terms by extracting and canceling out 2.

$$w=\frac{11}{2}$$

Now solve the equation w=\frac{3±19}{4} when ± is minus. Subtract 19 from 3.

$$w=-\frac{16}{4}$$

Divide -16 by 4.

$$w=-4$$

The equation is now solved.

$$w=\frac{11}{2} w=-4$$

Steps for Completing the Square

Use the distributive property to multiply w by 2w-3.

$$44=2w^{2}-3w$$

Swap sides so that all variable terms are on the left hand side.

$$2w^{2}-3w=44$$

Divide both sides by 2.

$$\frac{2w^{2}-3w}{2}=\frac{44}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$w^{2}-\frac{3}{2}w=\frac{44}{2}$$

Divide 44 by 2.

$$w^{2}-\frac{3}{2}w=22$$

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}-\frac{3}{2}w+\left(-\frac{3}{4}\right)^{2}=22+\left(-\frac{3}{4}\right)^{2}$$

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$w^{2}-\frac{3}{2}w+\frac{9}{16}=22+\frac{9}{16}$$

Add 22 to \frac{9}{16}.

$$w^{2}-\frac{3}{2}w+\frac{9}{16}=\frac{361}{16}$$

Factor w^{2}-\frac{3}{2}w+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(w-\frac{3}{4}\right)^{2}=\frac{361}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w-\frac{3}{4}\right)^{2}}=\sqrt{\frac{361}{16}}$$

Simplify.

$$w-\frac{3}{4}=\frac{19}{4} w-\frac{3}{4}=-\frac{19}{4}$$

Add \frac{3}{4} to both sides of the equation.

$$w=\frac{11}{2} w=-4$$