Solve (6-t)2t=9

$$( 6 - t ) 2 t = 9$$

Solve for t

$$t=\frac{3\sqrt{2}}{2}+3\approx 5.121320344$$

$$t=-\frac{3\sqrt{2}}{2}+3\approx 0.878679656$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 6-t by 2.

$$\left(12-2t\right)t=9$$

Use the distributive property to multiply 12-2t by t.

$$12t-2t^{2}=9$$

Subtract 9 from both sides.

$$12t-2t^{2}-9=0$$

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.

$$-2t^{2}+12t-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 12 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-12±\sqrt{12^{2}-4\left(-2\right)\left(-9\right)}}{2\left(-2\right)}$$

Square 12.

$$t=\frac{-12±\sqrt{144-4\left(-2\right)\left(-9\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$t=\frac{-12±\sqrt{144+8\left(-9\right)}}{2\left(-2\right)}$$

Multiply 8 times -9.

$$t=\frac{-12±\sqrt{144-72}}{2\left(-2\right)}$$

Add 144 to -72.

$$t=\frac{-12±\sqrt{72}}{2\left(-2\right)}$$

Take the square root of 72.

$$t=\frac{-12±6\sqrt{2}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$t=\frac{-12±6\sqrt{2}}{-4}$$

Now solve the equation t=\frac{-12±6\sqrt{2}}{-4} when ± is plus. Add -12 to 6\sqrt{2}.

$$t=\frac{6\sqrt{2}-12}{-4}$$

Divide -12+6\sqrt{2} by -4.

$$t=-\frac{3\sqrt{2}}{2}+3$$

Now solve the equation t=\frac{-12±6\sqrt{2}}{-4} when ± is minus. Subtract 6\sqrt{2} from -12.

$$t=\frac{-6\sqrt{2}-12}{-4}$$

Divide -12-6\sqrt{2} by -4.

$$t=\frac{3\sqrt{2}}{2}+3$$

The equation is now solved.

$$t=-\frac{3\sqrt{2}}{2}+3 t=\frac{3\sqrt{2}}{2}+3$$

Steps for Completing the Square

Use the distributive property to multiply 6-t by 2.

$$\left(12-2t\right)t=9$$

Use the distributive property to multiply 12-2t by t.

$$12t-2t^{2}=9$$

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.

$$-2t^{2}+12t=9$$

Divide both sides by -2.

$$\frac{-2t^{2}+12t}{-2}=\frac{9}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$t^{2}+\frac{12}{-2}t=\frac{9}{-2}$$

Divide 12 by -2.

$$t^{2}-6t=\frac{9}{-2}$$

Divide 9 by -2.

$$t^{2}-6t=-\frac{9}{2}$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-6t+\left(-3\right)^{2}=-\frac{9}{2}+\left(-3\right)^{2}$$

Square -3.

$$t^{2}-6t+9=-\frac{9}{2}+9$$

Add -\frac{9}{2} to 9.

$$t^{2}-6t+9=\frac{9}{2}$$

Factor t^{2}-6t+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-3\right)^{2}=\frac{9}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-3\right)^{2}}=\sqrt{\frac{9}{2}}$$

Simplify.

$$t-3=\frac{3\sqrt{2}}{2} t-3=-\frac{3\sqrt{2}}{2}$$

Add 3 to both sides of the equation.

$$t=\frac{3\sqrt{2}}{2}+3 t=-\frac{3\sqrt{2}}{2}+3$$