Solve 0=3t^2/2-11t+14

$$0 = \frac{ 3 { t }^{ 2 } }{ 2 } -11t+14$$

Solve for t

$$t = \frac{\sqrt{37} + 11}{3} \approx 5.694254177$$

$$t = \frac{11 - \sqrt{37}}{3} \approx 1.639079157$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$0=3t^{2}-22t+28$$

Swap sides so that all variable terms are on the left hand side.

$$3t^{2}-22t+28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -22 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 3\times 28}}{2\times 3}$$

Square -22.

$$t=\frac{-\left(-22\right)±\sqrt{484-4\times 3\times 28}}{2\times 3}$$

Multiply -4 times 3.

$$t=\frac{-\left(-22\right)±\sqrt{484-12\times 28}}{2\times 3}$$

Multiply -12 times 28.

$$t=\frac{-\left(-22\right)±\sqrt{484-336}}{2\times 3}$$

Add 484 to -336.

$$t=\frac{-\left(-22\right)±\sqrt{148}}{2\times 3}$$

Take the square root of 148.

$$t=\frac{-\left(-22\right)±2\sqrt{37}}{2\times 3}$$

The opposite of -22 is 22.

$$t=\frac{22±2\sqrt{37}}{2\times 3}$$

Multiply 2 times 3.

$$t=\frac{22±2\sqrt{37}}{6}$$

Now solve the equation t=\frac{22±2\sqrt{37}}{6} when ± is plus. Add 22 to 2\sqrt{37}.

$$t=\frac{2\sqrt{37}+22}{6}$$

Divide 22+2\sqrt{37} by 6.

$$t=\frac{\sqrt{37}+11}{3}$$

Now solve the equation t=\frac{22±2\sqrt{37}}{6} when ± is minus. Subtract 2\sqrt{37} from 22.

$$t=\frac{22-2\sqrt{37}}{6}$$

Divide 22-2\sqrt{37} by 6.

$$t=\frac{11-\sqrt{37}}{3}$$

The equation is now solved.

$$t=\frac{\sqrt{37}+11}{3} t=\frac{11-\sqrt{37}}{3}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$0=3t^{2}-22t+28$$

Swap sides so that all variable terms are on the left hand side.

$$3t^{2}-22t+28=0$$

Subtract 28 from both sides. Anything subtracted from zero gives its negation.

$$3t^{2}-22t=-28$$

Divide both sides by 3.

$$\frac{3t^{2}-22t}{3}=-\frac{28}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$t^{2}-\frac{22}{3}t=-\frac{28}{3}$$

Divide -\frac{22}{3}, the coefficient of the x term, by 2 to get -\frac{11}{3}. Then add the square of -\frac{11}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{22}{3}t+\left(-\frac{11}{3}\right)^{2}=-\frac{28}{3}+\left(-\frac{11}{3}\right)^{2}$$

Square -\frac{11}{3} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{22}{3}t+\frac{121}{9}=-\frac{28}{3}+\frac{121}{9}$$

Add -\frac{28}{3} to \frac{121}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{22}{3}t+\frac{121}{9}=\frac{37}{9}$$

Factor t^{2}-\frac{22}{3}t+\frac{121}{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}.

$$\left(t-\frac{11}{3}\right)^{2}=\frac{37}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{11}{3}\right)^{2}}=\sqrt{\frac{37}{9}}$$

Simplify.

$$t-\frac{11}{3}=\frac{\sqrt{37}}{3} t-\frac{11}{3}=-\frac{\sqrt{37}}{3}$$

Add \frac{11}{3} to both sides of the equation.

$$t=\frac{\sqrt{37}+11}{3} t=\frac{11-\sqrt{37}}{3}$$