Solve (12-5t)*400left(1+4/3tright)+9left(1+1/5tright)*200=6600left(1+3/5tright)

$$(12-5t) \times 400 \left( 1+ \frac{ 4 }{ 3 } t \right) +9 \left( 1+ \frac{ 1 }{ 5 } t \right) \times 200 = 6600 \left( 1+ \frac{ 3 }{ 5 } t \right)$$

Solve for t

$$t=\frac{3}{10}=0.3$$

$$t=0$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 12-5t by 400.

$$\left(4800-2000t\right)\left(1+\frac{4}{3}t\right)+9\left(1+\frac{1}{5}t\right)\times 200=6600\left(1+\frac{3}{5}t\right)$$

Use the distributive property to multiply 4800-2000t by 1+\frac{4}{3}t and combine like terms.

$$4800+4400t-\frac{8000}{3}t^{2}+9\left(1+\frac{1}{5}t\right)\times 200=6600\left(1+\frac{3}{5}t\right)$$

Multiply 9 and 200 to get 1800.

$$4800+4400t-\frac{8000}{3}t^{2}+1800\left(1+\frac{1}{5}t\right)=6600\left(1+\frac{3}{5}t\right)$$

Use the distributive property to multiply 1800 by 1+\frac{1}{5}t.

$$4800+4400t-\frac{8000}{3}t^{2}+1800+360t=6600\left(1+\frac{3}{5}t\right)$$

Add 4800 and 1800 to get 6600.

$$6600+4400t-\frac{8000}{3}t^{2}+360t=6600\left(1+\frac{3}{5}t\right)$$

Combine 4400t and 360t to get 4760t.

$$6600+4760t-\frac{8000}{3}t^{2}=6600\left(1+\frac{3}{5}t\right)$$

Use the distributive property to multiply 6600 by 1+\frac{3}{5}t.

$$6600+4760t-\frac{8000}{3}t^{2}=6600+3960t$$

Subtract 6600 from both sides.

$$6600+4760t-\frac{8000}{3}t^{2}-6600=3960t$$

Subtract 6600 from 6600 to get 0.

$$4760t-\frac{8000}{3}t^{2}=3960t$$

Subtract 3960t from both sides.

$$4760t-\frac{8000}{3}t^{2}-3960t=0$$

Combine 4760t and -3960t to get 800t.

$$800t-\frac{8000}{3}t^{2}=0$$

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.

$$-\frac{8000}{3}t^{2}+800t=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{8000}{3} for a, 800 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-800±\sqrt{800^{2}}}{2\left(-\frac{8000}{3}\right)}$$

Take the square root of 800^{2}.

$$t=\frac{-800±800}{2\left(-\frac{8000}{3}\right)}$$

Multiply 2 times -\frac{8000}{3}.

$$t=\frac{-800±800}{-\frac{16000}{3}}$$

Now solve the equation t=\frac{-800±800}{-\frac{16000}{3}} when ± is plus. Add -800 to 800.

$$t=\frac{0}{-\frac{16000}{3}}$$

Divide 0 by -\frac{16000}{3} by multiplying 0 by the reciprocal of -\frac{16000}{3}.

$$t=0$$

Now solve the equation t=\frac{-800±800}{-\frac{16000}{3}} when ± is minus. Subtract 800 from -800.

$$t=-\frac{1600}{-\frac{16000}{3}}$$

Divide -1600 by -\frac{16000}{3} by multiplying -1600 by the reciprocal of -\frac{16000}{3}.

$$t=\frac{3}{10}$$

The equation is now solved.

$$t=0 t=\frac{3}{10}$$

Steps for Completing the Square

Use the distributive property to multiply 12-5t by 400.

$$\left(4800-2000t\right)\left(1+\frac{4}{3}t\right)+9\left(1+\frac{1}{5}t\right)\times 200=6600\left(1+\frac{3}{5}t\right)$$

Use the distributive property to multiply 4800-2000t by 1+\frac{4}{3}t and combine like terms.

$$4800+4400t-\frac{8000}{3}t^{2}+9\left(1+\frac{1}{5}t\right)\times 200=6600\left(1+\frac{3}{5}t\right)$$

Multiply 9 and 200 to get 1800.

$$4800+4400t-\frac{8000}{3}t^{2}+1800\left(1+\frac{1}{5}t\right)=6600\left(1+\frac{3}{5}t\right)$$

Use the distributive property to multiply 1800 by 1+\frac{1}{5}t.

$$4800+4400t-\frac{8000}{3}t^{2}+1800+360t=6600\left(1+\frac{3}{5}t\right)$$

Add 4800 and 1800 to get 6600.

$$6600+4400t-\frac{8000}{3}t^{2}+360t=6600\left(1+\frac{3}{5}t\right)$$

Combine 4400t and 360t to get 4760t.

$$6600+4760t-\frac{8000}{3}t^{2}=6600\left(1+\frac{3}{5}t\right)$$

Use the distributive property to multiply 6600 by 1+\frac{3}{5}t.

$$6600+4760t-\frac{8000}{3}t^{2}=6600+3960t$$

Subtract 3960t from both sides.

$$6600+4760t-\frac{8000}{3}t^{2}-3960t=6600$$

Combine 4760t and -3960t to get 800t.

$$6600+800t-\frac{8000}{3}t^{2}=6600$$

Subtract 6600 from both sides.

$$800t-\frac{8000}{3}t^{2}=6600-6600$$

Subtract 6600 from 6600 to get 0.

$$800t-\frac{8000}{3}t^{2}=0$$

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.

$$-\frac{8000}{3}t^{2}+800t=0$$

Divide both sides of the equation by -\frac{8000}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{8000}{3}t^{2}+800t}{-\frac{8000}{3}}=\frac{0}{-\frac{8000}{3}}$$

Dividing by -\frac{8000}{3} undoes the multiplication by -\frac{8000}{3}.

$$t^{2}+\frac{800}{-\frac{8000}{3}}t=\frac{0}{-\frac{8000}{3}}$$

Divide 800 by -\frac{8000}{3} by multiplying 800 by the reciprocal of -\frac{8000}{3}.

$$t^{2}-\frac{3}{10}t=\frac{0}{-\frac{8000}{3}}$$

Divide 0 by -\frac{8000}{3} by multiplying 0 by the reciprocal of -\frac{8000}{3}.

$$t^{2}-\frac{3}{10}t=0$$

Divide -\frac{3}{10}, the coefficient of the x term, by 2 to get -\frac{3}{20}. Then add the square of -\frac{3}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{3}{10}t+\left(-\frac{3}{20}\right)^{2}=\left(-\frac{3}{20}\right)^{2}$$

Square -\frac{3}{20} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{3}{10}t+\frac{9}{400}=\frac{9}{400}$$

Factor t^{2}-\frac{3}{10}t+\frac{9}{400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{3}{20}\right)^{2}=\frac{9}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{3}{20}\right)^{2}}=\sqrt{\frac{9}{400}}$$

Simplify.

$$t-\frac{3}{20}=\frac{3}{20} t-\frac{3}{20}=-\frac{3}{20}$$

Add \frac{3}{20} to both sides of the equation.

$$t=\frac{3}{10} t=0$$