Solve (2-3/5t)^2=4(-2+4/5t)

$$( 2 - \frac { 3 } { 5 } t ) ^ { 2 } = 4 ( - 2 + \frac { 4 } { 5 } t )$$

Solve for t

$$t = \frac{10 \sqrt{22} + 70}{9} \approx 12.989350844$$

$$t = \frac{70 - 10 \sqrt{22}}{9} \approx 2.566204711$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2-\frac{3}{5}t\right)^{2}.

$$4-\frac{12}{5}t+\frac{9}{25}t^{2}=4\left(-2+\frac{4}{5}t\right)$$

Use the distributive property to multiply 4 by -2+\frac{4}{5}t.

$$4-\frac{12}{5}t+\frac{9}{25}t^{2}=-8+\frac{16}{5}t$$

Subtract -8 from both sides.

$$4-\frac{12}{5}t+\frac{9}{25}t^{2}-\left(-8\right)=\frac{16}{5}t$$

The opposite of -8 is 8.

$$4-\frac{12}{5}t+\frac{9}{25}t^{2}+8=\frac{16}{5}t$$

Subtract \frac{16}{5}t from both sides.

$$4-\frac{12}{5}t+\frac{9}{25}t^{2}+8-\frac{16}{5}t=0$$

Add 4 and 8 to get 12.

$$12-\frac{12}{5}t+\frac{9}{25}t^{2}-\frac{16}{5}t=0$$

Combine -\frac{12}{5}t and -\frac{16}{5}t to get -\frac{28}{5}t.

$$12-\frac{28}{5}t+\frac{9}{25}t^{2}=0$$

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.

$$\frac{9}{25}t^{2}-\frac{28}{5}t+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{9}{25} for a, -\frac{28}{5} for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-\frac{28}{5}\right)±\sqrt{\left(-\frac{28}{5}\right)^{2}-4\times \frac{9}{25}\times 12}}{2\times \frac{9}{25}}$$

Square -\frac{28}{5} by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-\left(-\frac{28}{5}\right)±\sqrt{\frac{784}{25}-4\times \frac{9}{25}\times 12}}{2\times \frac{9}{25}}$$

Multiply -4 times \frac{9}{25}.

$$t=\frac{-\left(-\frac{28}{5}\right)±\sqrt{\frac{784}{25}-\frac{36}{25}\times 12}}{2\times \frac{9}{25}}$$

Multiply -\frac{36}{25} times 12.

$$t=\frac{-\left(-\frac{28}{5}\right)±\sqrt{\frac{784-432}{25}}}{2\times \frac{9}{25}}$$

Add \frac{784}{25} to -\frac{432}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-\left(-\frac{28}{5}\right)±\sqrt{\frac{352}{25}}}{2\times \frac{9}{25}}$$

Take the square root of \frac{352}{25}.

$$t=\frac{-\left(-\frac{28}{5}\right)±\frac{4\sqrt{22}}{5}}{2\times \frac{9}{25}}$$

The opposite of -\frac{28}{5} is \frac{28}{5}.

$$t=\frac{\frac{28}{5}±\frac{4\sqrt{22}}{5}}{2\times \frac{9}{25}}$$

Multiply 2 times \frac{9}{25}.

$$t=\frac{\frac{28}{5}±\frac{4\sqrt{22}}{5}}{\frac{18}{25}}$$

Now solve the equation t=\frac{\frac{28}{5}±\frac{4\sqrt{22}}{5}}{\frac{18}{25}} when ± is plus. Add \frac{28}{5} to \frac{4\sqrt{22}}{5}.

$$t=\frac{4\sqrt{22}+28}{\frac{18}{25}\times 5}$$

Divide \frac{28+4\sqrt{22}}{5} by \frac{18}{25} by multiplying \frac{28+4\sqrt{22}}{5} by the reciprocal of \frac{18}{25}.

$$t=\frac{10\sqrt{22}+70}{9}$$

Now solve the equation t=\frac{\frac{28}{5}±\frac{4\sqrt{22}}{5}}{\frac{18}{25}} when ± is minus. Subtract \frac{4\sqrt{22}}{5} from \frac{28}{5}.

$$t=\frac{28-4\sqrt{22}}{\frac{18}{25}\times 5}$$

Divide \frac{28-4\sqrt{22}}{5} by \frac{18}{25} by multiplying \frac{28-4\sqrt{22}}{5} by the reciprocal of \frac{18}{25}.

$$t=\frac{70-10\sqrt{22}}{9}$$

The equation is now solved.

$$t=\frac{10\sqrt{22}+70}{9} t=\frac{70-10\sqrt{22}}{9}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2-\frac{3}{5}t\right)^{2}.

$$4-\frac{12}{5}t+\frac{9}{25}t^{2}=4\left(-2+\frac{4}{5}t\right)$$

Use the distributive property to multiply 4 by -2+\frac{4}{5}t.

$$4-\frac{12}{5}t+\frac{9}{25}t^{2}=-8+\frac{16}{5}t$$

Subtract \frac{16}{5}t from both sides.

$$4-\frac{12}{5}t+\frac{9}{25}t^{2}-\frac{16}{5}t=-8$$

Combine -\frac{12}{5}t and -\frac{16}{5}t to get -\frac{28}{5}t.

$$4-\frac{28}{5}t+\frac{9}{25}t^{2}=-8$$

Subtract 4 from both sides.

$$-\frac{28}{5}t+\frac{9}{25}t^{2}=-8-4$$

Subtract 4 from -8 to get -12.

$$-\frac{28}{5}t+\frac{9}{25}t^{2}=-12$$

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.

$$\frac{9}{25}t^{2}-\frac{28}{5}t=-12$$

Divide both sides of the equation by \frac{9}{25}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{9}{25}t^{2}-\frac{28}{5}t}{\frac{9}{25}}=-\frac{12}{\frac{9}{25}}$$

Dividing by \frac{9}{25} undoes the multiplication by \frac{9}{25}.

$$t^{2}+\left(-\frac{\frac{28}{5}}{\frac{9}{25}}\right)t=-\frac{12}{\frac{9}{25}}$$

Divide -\frac{28}{5} by \frac{9}{25} by multiplying -\frac{28}{5} by the reciprocal of \frac{9}{25}.

$$t^{2}-\frac{140}{9}t=-\frac{12}{\frac{9}{25}}$$

Divide -12 by \frac{9}{25} by multiplying -12 by the reciprocal of \frac{9}{25}.

$$t^{2}-\frac{140}{9}t=-\frac{100}{3}$$

Divide -\frac{140}{9}, the coefficient of the x term, by 2 to get -\frac{70}{9}. Then add the square of -\frac{70}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{140}{9}t+\left(-\frac{70}{9}\right)^{2}=-\frac{100}{3}+\left(-\frac{70}{9}\right)^{2}$$

Square -\frac{70}{9} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{140}{9}t+\frac{4900}{81}=-\frac{100}{3}+\frac{4900}{81}$$

Add -\frac{100}{3} to \frac{4900}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{140}{9}t+\frac{4900}{81}=\frac{2200}{81}$$

Factor t^{2}-\frac{140}{9}t+\frac{4900}{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}.

$$\left(t-\frac{70}{9}\right)^{2}=\frac{2200}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{70}{9}\right)^{2}}=\sqrt{\frac{2200}{81}}$$

Simplify.

$$t-\frac{70}{9}=\frac{10\sqrt{22}}{9} t-\frac{70}{9}=-\frac{10\sqrt{22}}{9}$$

Add \frac{70}{9} to both sides of the equation.

$$t=\frac{10\sqrt{22}+70}{9} t=\frac{70-10\sqrt{22}}{9}$$