Solve 3(w/6)^2-8(w/6)+4=0

$$3 ( \frac { w } { 6 } ) ^ { 2 } - 8 ( \frac { w } { 6 } ) + 4 = 0$$

Solve for w

$$w=4$$

$$w=12$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6.

$$6\left(3\times \left(\frac{w}{6}\right)^{2}-8\times \frac{w}{6}\right)+24=0$$

To raise \frac{w}{6} to a power, raise both numerator and denominator to the power and then divide.

$$6\left(3\times \frac{w^{2}}{6^{2}}-8\times \frac{w}{6}\right)+24=0$$

Express 3\times \frac{w^{2}}{6^{2}} as a single fraction.

$$6\left(\frac{3w^{2}}{6^{2}}-8\times \frac{w}{6}\right)+24=0$$

Express 8\times \frac{w}{6} as a single fraction.

$$6\left(\frac{3w^{2}}{6^{2}}-\frac{8w}{6}\right)+24=0$$

Divide 8w by 6 to get \frac{4}{3}w.

$$6\left(\frac{3w^{2}}{6^{2}}-\frac{4}{3}w\right)+24=0$$

Use the distributive property to multiply 6 by \frac{3w^{2}}{6^{2}}-\frac{4}{3}w.

$$6\times \frac{3w^{2}}{6^{2}}+6\left(-\frac{4}{3}w\right)+24=0$$

Calculate 6 to the power of 2 and get 36.

$$6\times \frac{3w^{2}}{36}+6\left(-\frac{4}{3}w\right)+24=0$$

Divide 3w^{2} by 36 to get \frac{1}{12}w^{2}.

$$6\times \frac{1}{12}w^{2}+6\left(-\frac{4}{3}w\right)+24=0$$

Multiply 6 and \frac{1}{12} to get \frac{1}{2}.

$$\frac{1}{2}w^{2}+6\left(-\frac{4}{3}w\right)+24=0$$

Multiply 6 and -1 to get -6.

$$\frac{1}{2}w^{2}-6\times \frac{4}{3}w+24=0$$

Multiply -6 and \frac{4}{3} to get -8.

$$\frac{1}{2}w^{2}-8w+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, -8 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times \frac{1}{2}\times 24}}{2\times \frac{1}{2}}$$

Square -8.

$$w=\frac{-\left(-8\right)±\sqrt{64-4\times \frac{1}{2}\times 24}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$w=\frac{-\left(-8\right)±\sqrt{64-2\times 24}}{2\times \frac{1}{2}}$$

Multiply -2 times 24.

$$w=\frac{-\left(-8\right)±\sqrt{64-48}}{2\times \frac{1}{2}}$$

Add 64 to -48.

$$w=\frac{-\left(-8\right)±\sqrt{16}}{2\times \frac{1}{2}}$$

Take the square root of 16.

$$w=\frac{-\left(-8\right)±4}{2\times \frac{1}{2}}$$

The opposite of -8 is 8.

$$w=\frac{8±4}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$w=\frac{8±4}{1}$$

Now solve the equation w=\frac{8±4}{1} when ± is plus. Add 8 to 4.

$$w=\frac{12}{1}$$

Divide 12 by 1.

$$w=12$$

Now solve the equation w=\frac{8±4}{1} when ± is minus. Subtract 4 from 8.

$$w=\frac{4}{1}$$

Divide 4 by 1.

$$w=4$$

The equation is now solved.

$$w=12 w=4$$

Steps for Completing the Square

Multiply both sides of the equation by 6.

$$6\left(3\times \left(\frac{w}{6}\right)^{2}-8\times \frac{w}{6}\right)+24=0$$

To raise \frac{w}{6} to a power, raise both numerator and denominator to the power and then divide.

$$6\left(3\times \frac{w^{2}}{6^{2}}-8\times \frac{w}{6}\right)+24=0$$

Express 3\times \frac{w^{2}}{6^{2}} as a single fraction.

$$6\left(\frac{3w^{2}}{6^{2}}-8\times \frac{w}{6}\right)+24=0$$

Express 8\times \frac{w}{6} as a single fraction.

$$6\left(\frac{3w^{2}}{6^{2}}-\frac{8w}{6}\right)+24=0$$

Divide 8w by 6 to get \frac{4}{3}w.

$$6\left(\frac{3w^{2}}{6^{2}}-\frac{4}{3}w\right)+24=0$$

Use the distributive property to multiply 6 by \frac{3w^{2}}{6^{2}}-\frac{4}{3}w.

$$6\times \frac{3w^{2}}{6^{2}}+6\left(-\frac{4}{3}w\right)+24=0$$

Calculate 6 to the power of 2 and get 36.

$$6\times \frac{3w^{2}}{36}+6\left(-\frac{4}{3}w\right)+24=0$$

Divide 3w^{2} by 36 to get \frac{1}{12}w^{2}.

$$6\times \frac{1}{12}w^{2}+6\left(-\frac{4}{3}w\right)+24=0$$

Multiply 6 and \frac{1}{12} to get \frac{1}{2}.

$$\frac{1}{2}w^{2}+6\left(-\frac{4}{3}w\right)+24=0$$

Subtract 24 from both sides. Anything subtracted from zero gives its negation.

$$\frac{1}{2}w^{2}+6\left(-\frac{4}{3}w\right)=-24$$

Multiply 6 and -1 to get -6.

$$\frac{1}{2}w^{2}-6\times \frac{4}{3}w=-24$$

Multiply -6 and \frac{4}{3} to get -8.

$$\frac{1}{2}w^{2}-8w=-24$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}w^{2}-8w}{\frac{1}{2}}=-\frac{24}{\frac{1}{2}}$$

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

$$w^{2}+\left(-\frac{8}{\frac{1}{2}}\right)w=-\frac{24}{\frac{1}{2}}$$

Divide -8 by \frac{1}{2} by multiplying -8 by the reciprocal of \frac{1}{2}.

$$w^{2}-16w=-\frac{24}{\frac{1}{2}}$$

Divide -24 by \frac{1}{2} by multiplying -24 by the reciprocal of \frac{1}{2}.

$$w^{2}-16w=-48$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}-16w+\left(-8\right)^{2}=-48+\left(-8\right)^{2}$$

Square -8.

$$w^{2}-16w+64=-48+64$$

Add -48 to 64.

$$w^{2}-16w+64=16$$

Factor w^{2}-16w+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}.

$$\left(w-8\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w-8\right)^{2}}=\sqrt{16}$$

Simplify.

$$w-8=4 w-8=-4$$

Add 8 to both sides of the equation.

$$w=12 w=4$$