Solve t/2+9-18/t=0

$$\frac { t } { 2 } + 9 - \frac { 18 } { t } = 0$$

Solve for t

$$t=3\sqrt{13}-9\approx 1.816653826$$

$$t=-3\sqrt{13}-9\approx -19.816653826$$

Steps Using the Quadratic Formula

Variable t cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2t, the least common multiple of 2,t.

$$tt+2t\times 9-2\times 18=0$$

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

$$t^{2}+2t\times 9-2\times 18=0$$

Multiply 2 and 9 to get 18.

$$t^{2}+18t-2\times 18=0$$

Multiply -2 and 18 to get -36.

$$t^{2}+18t-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 18 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-18±\sqrt{18^{2}-4\left(-36\right)}}{2}$$

Square 18.

$$t=\frac{-18±\sqrt{324-4\left(-36\right)}}{2}$$

Multiply -4 times -36.

$$t=\frac{-18±\sqrt{324+144}}{2}$$

Add 324 to 144.

$$t=\frac{-18±\sqrt{468}}{2}$$

Take the square root of 468.

$$t=\frac{-18±6\sqrt{13}}{2}$$

Now solve the equation t=\frac{-18±6\sqrt{13}}{2} when ± is plus. Add -18 to 6\sqrt{13}.

$$t=\frac{6\sqrt{13}-18}{2}$$

Divide -18+6\sqrt{13} by 2.

$$t=3\sqrt{13}-9$$

Now solve the equation t=\frac{-18±6\sqrt{13}}{2} when ± is minus. Subtract 6\sqrt{13} from -18.

$$t=\frac{-6\sqrt{13}-18}{2}$$

Divide -18-6\sqrt{13} by 2.

$$t=-3\sqrt{13}-9$$

The equation is now solved.

$$t=3\sqrt{13}-9 t=-3\sqrt{13}-9$$

Steps for Completing the Square

Variable t cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2t, the least common multiple of 2,t.

$$tt+2t\times 9-2\times 18=0$$

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

$$t^{2}+2t\times 9-2\times 18=0$$

Multiply 2 and 9 to get 18.

$$t^{2}+18t-2\times 18=0$$

Multiply -2 and 18 to get -36.

$$t^{2}+18t-36=0$$

Add 36 to both sides. Anything plus zero gives itself.

$$t^{2}+18t=36$$

Divide 18, the coefficient of the x term, by 2 to get 9. Then add the square of 9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+18t+9^{2}=36+9^{2}$$

Square 9.

$$t^{2}+18t+81=36+81$$

Add 36 to 81.

$$t^{2}+18t+81=117$$

Factor t^{2}+18t+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+9\right)^{2}=117$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+9\right)^{2}}=\sqrt{117}$$

Simplify.

$$t+9=3\sqrt{13} t+9=-3\sqrt{13}$$

Subtract 9 from both sides of the equation.

$$t=3\sqrt{13}-9 t=-3\sqrt{13}-9$$