Solve frac{20^{2}+(1.5t)^2-(12+1.5t)^2}{60t}=347(1.5t)^2-(2011.5t)^2/102t

$$\frac { 20 ^ { 2 } + ( 1.5 t ) ^ { 2 } - ( 12 + 1.5 t ) ^ { 2 } } { 60 t } = \frac { 347 ( 1.5 t ) ^ { 2 } - ( 2011.5 t ) ^ { 2 } } { 102 t }$$

Solve for t

$$t=\frac{i\times 2\sqrt{4890377879}}{13484505}+\frac{34}{4494835}\approx 0.000007564+0.010372088i$$

$$t=-\frac{i\times 2\sqrt{4890377879}}{13484505}+\frac{34}{4494835}\approx 0.000007564-0.010372088i$$

Steps Using the Quadratic Formula

Variable t cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 1020t, the least common multiple of 60t,102t.

$$17\left(20^{2}+\left(1.5t\right)^{2}-\left(12+1.5t\right)^{2}\right)=10\left(347\times \left(1.5t\right)^{2}-\left(2011.5t\right)^{2}\right)$$

Calculate 20 to the power of 2 and get 400.

$$17\left(400+\left(1.5t\right)^{2}-\left(12+1.5t\right)^{2}\right)=10\left(347\times \left(1.5t\right)^{2}-\left(2011.5t\right)^{2}\right)$$

Expand \left(1.5t\right)^{2}.

$$17\left(400+1.5^{2}t^{2}-\left(12+1.5t\right)^{2}\right)=10\left(347\times \left(1.5t\right)^{2}-\left(2011.5t\right)^{2}\right)$$

Calculate 1.5 to the power of 2 and get 2.25.

$$17\left(400+2.25t^{2}-\left(12+1.5t\right)^{2}\right)=10\left(347\times \left(1.5t\right)^{2}-\left(2011.5t\right)^{2}\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(12+1.5t\right)^{2}.

$$17\left(400+2.25t^{2}-\left(144+36t+2.25t^{2}\right)\right)=10\left(347\times \left(1.5t\right)^{2}-\left(2011.5t\right)^{2}\right)$$

To find the opposite of 144+36t+2.25t^{2}, find the opposite of each term.

$$17\left(400+2.25t^{2}-144-36t-2.25t^{2}\right)=10\left(347\times \left(1.5t\right)^{2}-\left(2011.5t\right)^{2}\right)$$

Subtract 144 from 400 to get 256.

$$17\left(256+2.25t^{2}-36t-2.25t^{2}\right)=10\left(347\times \left(1.5t\right)^{2}-\left(2011.5t\right)^{2}\right)$$

Combine 2.25t^{2} and -2.25t^{2} to get 0.

$$17\left(256-36t\right)=10\left(347\times \left(1.5t\right)^{2}-\left(2011.5t\right)^{2}\right)$$

Use the distributive property to multiply 17 by 256-36t.

$$4352-612t=10\left(347\times \left(1.5t\right)^{2}-\left(2011.5t\right)^{2}\right)$$

Expand \left(1.5t\right)^{2}.

$$4352-612t=10\left(347\times 1.5^{2}t^{2}-\left(2011.5t\right)^{2}\right)$$

Calculate 1.5 to the power of 2 and get 2.25.

$$4352-612t=10\left(347\times 2.25t^{2}-\left(2011.5t\right)^{2}\right)$$

Multiply 347 and 2.25 to get 780.75.

$$4352-612t=10\left(780.75t^{2}-\left(2011.5t\right)^{2}\right)$$

Expand \left(2011.5t\right)^{2}.

$$4352-612t=10\left(780.75t^{2}-2011.5^{2}t^{2}\right)$$

Calculate 2011.5 to the power of 2 and get 4046132.25.

$$4352-612t=10\left(780.75t^{2}-4046132.25t^{2}\right)$$

Combine 780.75t^{2} and -4046132.25t^{2} to get -4045351.5t^{2}.

$$4352-612t=10\left(-4045351.5\right)t^{2}$$

Multiply 10 and -4045351.5 to get -40453515.

$$4352-612t=-40453515t^{2}$$

Add 40453515t^{2} to both sides.

$$4352-612t+40453515t^{2}=0$$

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.

$$40453515t^{2}-612t+4352=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 40453515 for a, -612 for b, and 4352 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-612\right)±\sqrt{\left(-612\right)^{2}-4\times 40453515\times 4352}}{2\times 40453515}$$

Square -612.

$$t=\frac{-\left(-612\right)±\sqrt{374544-4\times 40453515\times 4352}}{2\times 40453515}$$

Multiply -4 times 40453515.

$$t=\frac{-\left(-612\right)±\sqrt{374544-161814060\times 4352}}{2\times 40453515}$$

Multiply -161814060 times 4352.

$$t=\frac{-\left(-612\right)±\sqrt{374544-704214789120}}{2\times 40453515}$$

Add 374544 to -704214789120.

$$t=\frac{-\left(-612\right)±\sqrt{-704214414576}}{2\times 40453515}$$

Take the square root of -704214414576.

$$t=\frac{-\left(-612\right)±12\sqrt{4890377879}i}{2\times 40453515}$$

The opposite of -612 is 612.

$$t=\frac{612±12\sqrt{4890377879}i}{2\times 40453515}$$

Multiply 2 times 40453515.

$$t=\frac{612±12\sqrt{4890377879}i}{80907030}$$

Now solve the equation t=\frac{612±12\sqrt{4890377879}i}{80907030} when ± is plus. Add 612 to 12i\sqrt{4890377879}.

$$t=\frac{612+12\sqrt{4890377879}i}{80907030}$$

Divide 612+12i\sqrt{4890377879} by 80907030.

$$t=\frac{2\sqrt{4890377879}i}{13484505}+\frac{34}{4494835}$$

Now solve the equation t=\frac{612±12\sqrt{4890377879}i}{80907030} when ± is minus. Subtract 12i\sqrt{4890377879} from 612.

$$t=\frac{-12\sqrt{4890377879}i+612}{80907030}$$

Divide 612-12i\sqrt{4890377879} by 80907030.

$$t=-\frac{2\sqrt{4890377879}i}{13484505}+\frac{34}{4494835}$$

The equation is now solved.

$$t=\frac{2\sqrt{4890377879}i}{13484505}+\frac{34}{4494835} t=-\frac{2\sqrt{4890377879}i}{13484505}+\frac{34}{4494835}$$

Steps for Completing the Square

Variable t cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 1020t, the least common multiple of 60t,102t.

$$17\left(20^{2}+\left(1.5t\right)^{2}-\left(12+1.5t\right)^{2}\right)=10\left(347\times \left(1.5t\right)^{2}-\left(2011.5t\right)^{2}\right)$$

Calculate 20 to the power of 2 and get 400.

$$17\left(400+\left(1.5t\right)^{2}-\left(12+1.5t\right)^{2}\right)=10\left(347\times \left(1.5t\right)^{2}-\left(2011.5t\right)^{2}\right)$$

Expand \left(1.5t\right)^{2}.

$$17\left(400+1.5^{2}t^{2}-\left(12+1.5t\right)^{2}\right)=10\left(347\times \left(1.5t\right)^{2}-\left(2011.5t\right)^{2}\right)$$

Calculate 1.5 to the power of 2 and get 2.25.

$$17\left(400+2.25t^{2}-\left(12+1.5t\right)^{2}\right)=10\left(347\times \left(1.5t\right)^{2}-\left(2011.5t\right)^{2}\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(12+1.5t\right)^{2}.

$$17\left(400+2.25t^{2}-\left(144+36t+2.25t^{2}\right)\right)=10\left(347\times \left(1.5t\right)^{2}-\left(2011.5t\right)^{2}\right)$$

To find the opposite of 144+36t+2.25t^{2}, find the opposite of each term.

$$17\left(400+2.25t^{2}-144-36t-2.25t^{2}\right)=10\left(347\times \left(1.5t\right)^{2}-\left(2011.5t\right)^{2}\right)$$

Subtract 144 from 400 to get 256.

$$17\left(256+2.25t^{2}-36t-2.25t^{2}\right)=10\left(347\times \left(1.5t\right)^{2}-\left(2011.5t\right)^{2}\right)$$

Combine 2.25t^{2} and -2.25t^{2} to get 0.

$$17\left(256-36t\right)=10\left(347\times \left(1.5t\right)^{2}-\left(2011.5t\right)^{2}\right)$$

Use the distributive property to multiply 17 by 256-36t.

$$4352-612t=10\left(347\times \left(1.5t\right)^{2}-\left(2011.5t\right)^{2}\right)$$

Expand \left(1.5t\right)^{2}.

$$4352-612t=10\left(347\times 1.5^{2}t^{2}-\left(2011.5t\right)^{2}\right)$$

Calculate 1.5 to the power of 2 and get 2.25.

$$4352-612t=10\left(347\times 2.25t^{2}-\left(2011.5t\right)^{2}\right)$$

Multiply 347 and 2.25 to get 780.75.

$$4352-612t=10\left(780.75t^{2}-\left(2011.5t\right)^{2}\right)$$

Expand \left(2011.5t\right)^{2}.

$$4352-612t=10\left(780.75t^{2}-2011.5^{2}t^{2}\right)$$

Calculate 2011.5 to the power of 2 and get 4046132.25.

$$4352-612t=10\left(780.75t^{2}-4046132.25t^{2}\right)$$

Combine 780.75t^{2} and -4046132.25t^{2} to get -4045351.5t^{2}.

$$4352-612t=10\left(-4045351.5\right)t^{2}$$

Multiply 10 and -4045351.5 to get -40453515.

$$4352-612t=-40453515t^{2}$$

Add 40453515t^{2} to both sides.

$$4352-612t+40453515t^{2}=0$$

Subtract 4352 from both sides. Anything subtracted from zero gives its negation.

$$-612t+40453515t^{2}=-4352$$

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.

$$40453515t^{2}-612t=-4352$$

Divide both sides by 40453515.

$$\frac{40453515t^{2}-612t}{40453515}=-\frac{4352}{40453515}$$

Dividing by 40453515 undoes the multiplication by 40453515.

$$t^{2}+\left(-\frac{612}{40453515}\right)t=-\frac{4352}{40453515}$$

Reduce the fraction \frac{-612}{40453515} to lowest terms by extracting and canceling out 9.

$$t^{2}-\frac{68}{4494835}t=-\frac{4352}{40453515}$$

Divide -\frac{68}{4494835}, the coefficient of the x term, by 2 to get -\frac{34}{4494835}. Then add the square of -\frac{34}{4494835} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{68}{4494835}t+\left(-\frac{34}{4494835}\right)^{2}=-\frac{4352}{40453515}+\left(-\frac{34}{4494835}\right)^{2}$$

Square -\frac{34}{4494835} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{68}{4494835}t+\frac{1156}{20203541677225}=-\frac{4352}{40453515}+\frac{1156}{20203541677225}$$

Add -\frac{4352}{40453515} to \frac{1156}{20203541677225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{68}{4494835}t+\frac{1156}{20203541677225}=-\frac{19561511516}{181831875095025}$$

Factor t^{2}-\frac{68}{4494835}t+\frac{1156}{20203541677225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{34}{4494835}\right)^{2}=-\frac{19561511516}{181831875095025}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{34}{4494835}\right)^{2}}=\sqrt{-\frac{19561511516}{181831875095025}}$$

Simplify.

$$t-\frac{34}{4494835}=\frac{2\sqrt{4890377879}i}{13484505} t-\frac{34}{4494835}=-\frac{2\sqrt{4890377879}i}{13484505}$$

Add \frac{34}{4494835} to both sides of the equation.

$$t=\frac{2\sqrt{4890377879}i}{13484505}+\frac{34}{4494835} t=-\frac{2\sqrt{4890377879}i}{13484505}+\frac{34}{4494835}$$