Solve 1/2{left(t-4/3right)}^2-1/2left(t-4/3right)=frac{2}{3}t

$$\frac{ 1 }{ 2 } { \left(t- \frac{ 4 }{ 3 } \right) }^{ 2 } - \frac{ 1 }{ 2 } \left( t- \frac{ 4 }{ 3 } \right) = \frac{ 2 }{ 3 } t$$

Solve for t

$$t=\frac{\sqrt{113}}{6}+\frac{5}{2}\approx 4.271690969$$

$$t=-\frac{\sqrt{113}}{6}+\frac{5}{2}\approx 0.728309031$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(t-\frac{4}{3}\right)^{2}.

$$\frac{1}{2}\left(t^{2}-\frac{8}{3}t+\frac{16}{9}\right)-\frac{1}{2}\left(t-\frac{4}{3}\right)=\frac{2}{3}t$$

Use the distributive property to multiply \frac{1}{2} by t^{2}-\frac{8}{3}t+\frac{16}{9}.

$$\frac{1}{2}t^{2}-\frac{4}{3}t+\frac{8}{9}-\frac{1}{2}\left(t-\frac{4}{3}\right)=\frac{2}{3}t$$

Use the distributive property to multiply -\frac{1}{2} by t-\frac{4}{3}.

$$\frac{1}{2}t^{2}-\frac{4}{3}t+\frac{8}{9}-\frac{1}{2}t+\frac{2}{3}=\frac{2}{3}t$$

Combine -\frac{4}{3}t and -\frac{1}{2}t to get -\frac{11}{6}t.

$$\frac{1}{2}t^{2}-\frac{11}{6}t+\frac{8}{9}+\frac{2}{3}=\frac{2}{3}t$$

Add \frac{8}{9} and \frac{2}{3} to get \frac{14}{9}.

$$\frac{1}{2}t^{2}-\frac{11}{6}t+\frac{14}{9}=\frac{2}{3}t$$

Subtract \frac{2}{3}t from both sides.

$$\frac{1}{2}t^{2}-\frac{11}{6}t+\frac{14}{9}-\frac{2}{3}t=0$$

Combine -\frac{11}{6}t and -\frac{2}{3}t to get -\frac{5}{2}t.

$$\frac{1}{2}t^{2}-\frac{5}{2}t+\frac{14}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, -\frac{5}{2} for b, and \frac{14}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-\frac{5}{2}\right)±\sqrt{\left(-\frac{5}{2}\right)^{2}-4\times \frac{1}{2}\times \frac{14}{9}}}{2\times \frac{1}{2}}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-\left(-\frac{5}{2}\right)±\sqrt{\frac{25}{4}-4\times \frac{1}{2}\times \frac{14}{9}}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$t=\frac{-\left(-\frac{5}{2}\right)±\sqrt{\frac{25}{4}-2\times \frac{14}{9}}}{2\times \frac{1}{2}}$$

Multiply -2 times \frac{14}{9}.

$$t=\frac{-\left(-\frac{5}{2}\right)±\sqrt{\frac{25}{4}-\frac{28}{9}}}{2\times \frac{1}{2}}$$

Add \frac{25}{4} to -\frac{28}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-\left(-\frac{5}{2}\right)±\sqrt{\frac{113}{36}}}{2\times \frac{1}{2}}$$

Take the square root of \frac{113}{36}.

$$t=\frac{-\left(-\frac{5}{2}\right)±\frac{\sqrt{113}}{6}}{2\times \frac{1}{2}}$$

The opposite of -\frac{5}{2} is \frac{5}{2}.

$$t=\frac{\frac{5}{2}±\frac{\sqrt{113}}{6}}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$t=\frac{\frac{5}{2}±\frac{\sqrt{113}}{6}}{1}$$

Now solve the equation t=\frac{\frac{5}{2}±\frac{\sqrt{113}}{6}}{1} when ± is plus. Add \frac{5}{2} to \frac{\sqrt{113}}{6}.

$$t=\frac{\frac{\sqrt{113}}{6}+\frac{5}{2}}{1}$$

Divide \frac{5}{2}+\frac{\sqrt{113}}{6} by 1.

$$t=\frac{\sqrt{113}}{6}+\frac{5}{2}$$

Now solve the equation t=\frac{\frac{5}{2}±\frac{\sqrt{113}}{6}}{1} when ± is minus. Subtract \frac{\sqrt{113}}{6} from \frac{5}{2}.

$$t=\frac{-\frac{\sqrt{113}}{6}+\frac{5}{2}}{1}$$

Divide \frac{5}{2}-\frac{\sqrt{113}}{6} by 1.

$$t=-\frac{\sqrt{113}}{6}+\frac{5}{2}$$

The equation is now solved.

$$t=\frac{\sqrt{113}}{6}+\frac{5}{2} t=-\frac{\sqrt{113}}{6}+\frac{5}{2}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(t-\frac{4}{3}\right)^{2}.

$$\frac{1}{2}\left(t^{2}-\frac{8}{3}t+\frac{16}{9}\right)-\frac{1}{2}\left(t-\frac{4}{3}\right)=\frac{2}{3}t$$

Use the distributive property to multiply \frac{1}{2} by t^{2}-\frac{8}{3}t+\frac{16}{9}.

$$\frac{1}{2}t^{2}-\frac{4}{3}t+\frac{8}{9}-\frac{1}{2}\left(t-\frac{4}{3}\right)=\frac{2}{3}t$$

Use the distributive property to multiply -\frac{1}{2} by t-\frac{4}{3}.

$$\frac{1}{2}t^{2}-\frac{4}{3}t+\frac{8}{9}-\frac{1}{2}t+\frac{2}{3}=\frac{2}{3}t$$

Combine -\frac{4}{3}t and -\frac{1}{2}t to get -\frac{11}{6}t.

$$\frac{1}{2}t^{2}-\frac{11}{6}t+\frac{8}{9}+\frac{2}{3}=\frac{2}{3}t$$

Add \frac{8}{9} and \frac{2}{3} to get \frac{14}{9}.

$$\frac{1}{2}t^{2}-\frac{11}{6}t+\frac{14}{9}=\frac{2}{3}t$$

Subtract \frac{2}{3}t from both sides.

$$\frac{1}{2}t^{2}-\frac{11}{6}t+\frac{14}{9}-\frac{2}{3}t=0$$

Combine -\frac{11}{6}t and -\frac{2}{3}t to get -\frac{5}{2}t.

$$\frac{1}{2}t^{2}-\frac{5}{2}t+\frac{14}{9}=0$$

Subtract \frac{14}{9} from both sides. Anything subtracted from zero gives its negation.

$$\frac{1}{2}t^{2}-\frac{5}{2}t=-\frac{14}{9}$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}t^{2}-\frac{5}{2}t}{\frac{1}{2}}=-\frac{\frac{14}{9}}{\frac{1}{2}}$$

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

$$t^{2}+\left(-\frac{\frac{5}{2}}{\frac{1}{2}}\right)t=-\frac{\frac{14}{9}}{\frac{1}{2}}$$

Divide -\frac{5}{2} by \frac{1}{2} by multiplying -\frac{5}{2} by the reciprocal of \frac{1}{2}.

$$t^{2}-5t=-\frac{\frac{14}{9}}{\frac{1}{2}}$$

Divide -\frac{14}{9} by \frac{1}{2} by multiplying -\frac{14}{9} by the reciprocal of \frac{1}{2}.

$$t^{2}-5t=-\frac{28}{9}$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-5t+\left(-\frac{5}{2}\right)^{2}=-\frac{28}{9}+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-5t+\frac{25}{4}=-\frac{28}{9}+\frac{25}{4}$$

Add -\frac{28}{9} to \frac{25}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-5t+\frac{25}{4}=\frac{113}{36}$$

Factor t^{2}-5t+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{5}{2}\right)^{2}=\frac{113}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{5}{2}\right)^{2}}=\sqrt{\frac{113}{36}}$$

Simplify.

$$t-\frac{5}{2}=\frac{\sqrt{113}}{6} t-\frac{5}{2}=-\frac{\sqrt{113}}{6}$$

Add \frac{5}{2} to both sides of the equation.

$$t=\frac{\sqrt{113}}{6}+\frac{5}{2} t=-\frac{\sqrt{113}}{6}+\frac{5}{2}$$