Solve (3+2w)w=54

$$( 3 + 2 w ) w = 54$$

Solve for w

$$w=-6$$

$$w = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

Steps Using the Quadratic Formula

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

$$3w+2w^{2}=54$$

Subtract 54 from both sides.

$$3w+2w^{2}-54=0$$

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.

$$2w^{2}+3w-54=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 3 for b, and -54 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-3±\sqrt{3^{2}-4\times 2\left(-54\right)}}{2\times 2}$$

Square 3.

$$w=\frac{-3±\sqrt{9-4\times 2\left(-54\right)}}{2\times 2}$$

Multiply -4 times 2.

$$w=\frac{-3±\sqrt{9-8\left(-54\right)}}{2\times 2}$$

Multiply -8 times -54.

$$w=\frac{-3±\sqrt{9+432}}{2\times 2}$$

Add 9 to 432.

$$w=\frac{-3±\sqrt{441}}{2\times 2}$$

Take the square root of 441.

$$w=\frac{-3±21}{2\times 2}$$

Multiply 2 times 2.

$$w=\frac{-3±21}{4}$$

Now solve the equation w=\frac{-3±21}{4} when ± is plus. Add -3 to 21.

$$w=\frac{18}{4}$$

Reduce the fraction \frac{18}{4} to lowest terms by extracting and canceling out 2.

$$w=\frac{9}{2}$$

Now solve the equation w=\frac{-3±21}{4} when ± is minus. Subtract 21 from -3.

$$w=-\frac{24}{4}$$

Divide -24 by 4.

$$w=-6$$

The equation is now solved.

$$w=\frac{9}{2} w=-6$$

Steps for Completing the Square

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

$$3w+2w^{2}=54$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$2w^{2}+3w=54$$

Divide both sides by 2.

$$\frac{2w^{2}+3w}{2}=\frac{54}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$w^{2}+\frac{3}{2}w=\frac{54}{2}$$

Divide 54 by 2.

$$w^{2}+\frac{3}{2}w=27$$

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.

$$w^{2}+\frac{3}{2}w+\left(\frac{3}{4}\right)^{2}=27+\left(\frac{3}{4}\right)^{2}$$

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

$$w^{2}+\frac{3}{2}w+\frac{9}{16}=27+\frac{9}{16}$$

Add 27 to \frac{9}{16}.

$$w^{2}+\frac{3}{2}w+\frac{9}{16}=\frac{441}{16}$$

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}.

$$\left(w+\frac{3}{4}\right)^{2}=\frac{441}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w+\frac{3}{4}\right)^{2}}=\sqrt{\frac{441}{16}}$$

Simplify.

$$w+\frac{3}{4}=\frac{21}{4} w+\frac{3}{4}=-\frac{21}{4}$$

Subtract \frac{3}{4} from both sides of the equation.

$$w=\frac{9}{2} w=-6$$