Solve 2xdiv3xdiv9=8(6-x)

$$2x \div 3x \div 9=8(6-x)$$

Solve for x

$$x=18\sqrt{11}-54\approx 5.699246226$$

$$x=-18\sqrt{11}-54\approx -113.699246226$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 9.

$$\frac{2x}{3}x=72\left(6-x\right)$$

Express \frac{2x}{3}x as a single fraction.

$$\frac{2xx}{3}=72\left(6-x\right)$$

Use the distributive property to multiply 72 by 6-x.

$$\frac{2xx}{3}=432-72x$$

Multiply x and x to get x^{2}.

$$\frac{2x^{2}}{3}=432-72x$$

Subtract 432 from both sides.

$$\frac{2x^{2}}{3}-432=-72x$$

Add 72x to both sides.

$$\frac{2x^{2}}{3}-432+72x=0$$

Multiply both sides of the equation by 3.

$$2x^{2}-1296+216x=0$$

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.

$$2x^{2}+216x-1296=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 216 for b, and -1296 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-216±\sqrt{216^{2}-4\times 2\left(-1296\right)}}{2\times 2}$$

Square 216.

$$x=\frac{-216±\sqrt{46656-4\times 2\left(-1296\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-216±\sqrt{46656-8\left(-1296\right)}}{2\times 2}$$

Multiply -8 times -1296.

$$x=\frac{-216±\sqrt{46656+10368}}{2\times 2}$$

Add 46656 to 10368.

$$x=\frac{-216±\sqrt{57024}}{2\times 2}$$

Take the square root of 57024.

$$x=\frac{-216±72\sqrt{11}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-216±72\sqrt{11}}{4}$$

Now solve the equation x=\frac{-216±72\sqrt{11}}{4} when ± is plus. Add -216 to 72\sqrt{11}.

$$x=\frac{72\sqrt{11}-216}{4}$$

Divide -216+72\sqrt{11} by 4.

$$x=18\sqrt{11}-54$$

Now solve the equation x=\frac{-216±72\sqrt{11}}{4} when ± is minus. Subtract 72\sqrt{11} from -216.

$$x=\frac{-72\sqrt{11}-216}{4}$$

Divide -216-72\sqrt{11} by 4.

$$x=-18\sqrt{11}-54$$

The equation is now solved.

$$x=18\sqrt{11}-54 x=-18\sqrt{11}-54$$

Steps for Completing the Square

Multiply both sides of the equation by 9.

$$\frac{2x}{3}x=72\left(6-x\right)$$

Express \frac{2x}{3}x as a single fraction.

$$\frac{2xx}{3}=72\left(6-x\right)$$

Use the distributive property to multiply 72 by 6-x.

$$\frac{2xx}{3}=432-72x$$

Multiply x and x to get x^{2}.

$$\frac{2x^{2}}{3}=432-72x$$

Add 72x to both sides.

$$\frac{2x^{2}}{3}+72x=432$$

Multiply both sides of the equation by 3.

$$2x^{2}+216x=1296$$

Divide both sides by 2.

$$\frac{2x^{2}+216x}{2}=\frac{1296}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{216}{2}x=\frac{1296}{2}$$

Divide 216 by 2.

$$x^{2}+108x=\frac{1296}{2}$$

Divide 1296 by 2.

$$x^{2}+108x=648$$

Divide 108, the coefficient of the x term, by 2 to get 54. Then add the square of 54 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+108x+54^{2}=648+54^{2}$$

Square 54.

$$x^{2}+108x+2916=648+2916$$

Add 648 to 2916.

$$x^{2}+108x+2916=3564$$

Factor x^{2}+108x+2916. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+54\right)^{2}=3564$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+54\right)^{2}}=\sqrt{3564}$$

Simplify.

$$x+54=18\sqrt{11} x+54=-18\sqrt{11}$$

Subtract 54 from both sides of the equation.

$$x=18\sqrt{11}-54 x=-18\sqrt{11}-54$$