Solve 3t(t+3)=2t-5

$$3 t ( t + 3 ) = 2 t - 5$$

Solve for t

$$t=\frac{-7+\sqrt{11}i}{6}\approx -1.166666667+0.552770798i$$

$$t=\frac{-\sqrt{11}i-7}{6}\approx -1.166666667-0.552770798i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3t by t+3.

$$3t^{2}+9t=2t-5$$

Subtract 2t from both sides.

$$3t^{2}+9t-2t=-5$$

Combine 9t and -2t to get 7t.

$$3t^{2}+7t=-5$$

Add 5 to both sides.

$$3t^{2}+7t+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 7 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-7±\sqrt{7^{2}-4\times 3\times 5}}{2\times 3}$$

Square 7.

$$t=\frac{-7±\sqrt{49-4\times 3\times 5}}{2\times 3}$$

Multiply -4 times 3.

$$t=\frac{-7±\sqrt{49-12\times 5}}{2\times 3}$$

Multiply -12 times 5.

$$t=\frac{-7±\sqrt{49-60}}{2\times 3}$$

Add 49 to -60.

$$t=\frac{-7±\sqrt{-11}}{2\times 3}$$

Take the square root of -11.

$$t=\frac{-7±\sqrt{11}i}{2\times 3}$$

Multiply 2 times 3.

$$t=\frac{-7±\sqrt{11}i}{6}$$

Now solve the equation t=\frac{-7±\sqrt{11}i}{6} when ± is plus. Add -7 to i\sqrt{11}.

$$t=\frac{-7+\sqrt{11}i}{6}$$

Now solve the equation t=\frac{-7±\sqrt{11}i}{6} when ± is minus. Subtract i\sqrt{11} from -7.

$$t=\frac{-\sqrt{11}i-7}{6}$$

The equation is now solved.

$$t=\frac{-7+\sqrt{11}i}{6} t=\frac{-\sqrt{11}i-7}{6}$$

Steps for Completing the Square

Use the distributive property to multiply 3t by t+3.

$$3t^{2}+9t=2t-5$$

Subtract 2t from both sides.

$$3t^{2}+9t-2t=-5$$

Combine 9t and -2t to get 7t.

$$3t^{2}+7t=-5$$

Divide both sides by 3.

$$\frac{3t^{2}+7t}{3}=-\frac{5}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$t^{2}+\frac{7}{3}t=-\frac{5}{3}$$

Divide \frac{7}{3}, the coefficient of the x term, by 2 to get \frac{7}{6}. Then add the square of \frac{7}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{7}{3}t+\left(\frac{7}{6}\right)^{2}=-\frac{5}{3}+\left(\frac{7}{6}\right)^{2}$$

Square \frac{7}{6} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{7}{3}t+\frac{49}{36}=-\frac{5}{3}+\frac{49}{36}$$

Add -\frac{5}{3} to \frac{49}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}+\frac{7}{3}t+\frac{49}{36}=-\frac{11}{36}$$

Factor t^{2}+\frac{7}{3}t+\frac{49}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{7}{6}\right)^{2}=-\frac{11}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{7}{6}\right)^{2}}=\sqrt{-\frac{11}{36}}$$

Simplify.

$$t+\frac{7}{6}=\frac{\sqrt{11}i}{6} t+\frac{7}{6}=-\frac{\sqrt{11}i}{6}$$

Subtract \frac{7}{6} from both sides of the equation.

$$t=\frac{-7+\sqrt{11}i}{6} t=\frac{-\sqrt{11}i-7}{6}$$