Solve 1.684*0.5+1.684*0.5/1.684-1.684*0.5=0.545t+0.5+t/1+frac{4{2}t}

$$1.684 \cdot 0.5+ \frac{ 1.684 \cdot 0.5 }{ 1.684-1.684 \cdot 0.5 } = 0.545t+ \frac{ 0.5+t }{ 1+ \frac{ 4 }{ 2 } t }$$

Solve for t

$$t = \frac{1342}{545} = 2\frac{252}{545} \approx 2.462385321$$

Steps Using the Quadratic Formula

Multiply 1.684 and 0.5 to get 0.842.

$$0.842+\frac{1.684\times 0.5}{1.684-1.684\times 0.5}=0.545t+\frac{0.5+t}{1+\frac{4}{2}t}$$

Multiply 1.684 and 0.5 to get 0.842.

$$0.842+\frac{0.842}{1.684-1.684\times 0.5}=0.545t+\frac{0.5+t}{1+\frac{4}{2}t}$$

Multiply 1.684 and 0.5 to get 0.842.

$$0.842+\frac{0.842}{1.684-0.842}=0.545t+\frac{0.5+t}{1+\frac{4}{2}t}$$

Subtract 0.842 from 1.684 to get 0.842.

$$0.842+\frac{0.842}{0.842}=0.545t+\frac{0.5+t}{1+\frac{4}{2}t}$$

Divide 0.842 by 0.842 to get 1.

$$0.842+1=0.545t+\frac{0.5+t}{1+\frac{4}{2}t}$$

Add 0.842 and 1 to get 1.842.

$$1.842=0.545t+\frac{0.5+t}{1+\frac{4}{2}t}$$

Divide 4 by 2 to get 2.

$$1.842=0.545t+\frac{0.5+t}{1+2t}$$

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

$$0.545t+\frac{0.5+t}{1+2t}=1.842$$

Subtract 1.842 from both sides.

$$0.545t+\frac{0.5+t}{1+2t}-1.842=0$$

Variable t cannot be equal to -\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 2t+1.

$$0.545t\left(2t+1\right)+0.5+t+\left(2t+1\right)\left(-1.842\right)=0$$

Reorder the terms.

$$0.545t\left(2t+1\right)-1.842\left(2t+1\right)+t+0.5=0$$

Use the distributive property to multiply 0.545t by 2t+1.

$$1.09t^{2}+0.545t-1.842\left(2t+1\right)+t+0.5=0$$

Use the distributive property to multiply -1.842 by 2t+1.

$$1.09t^{2}+0.545t-3.684t-1.842+t+0.5=0$$

Combine 0.545t and -3.684t to get -3.139t.

$$1.09t^{2}-3.139t-1.842+t+0.5=0$$

Combine -3.139t and t to get -2.139t.

$$1.09t^{2}-2.139t-1.842+0.5=0$$

Add -1.842 and 0.5 to get -1.342.

$$1.09t^{2}-2.139t-1.342=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1.09 for a, -2.139 for b, and -1.342 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-2.139\right)±\sqrt{\left(-2.139\right)^{2}-4\times 1.09\left(-1.342\right)}}{2\times 1.09}$$

Square -2.139 by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-\left(-2.139\right)±\sqrt{4.575321-4\times 1.09\left(-1.342\right)}}{2\times 1.09}$$

Multiply -4 times 1.09.

$$t=\frac{-\left(-2.139\right)±\sqrt{4.575321-4.36\left(-1.342\right)}}{2\times 1.09}$$

Multiply -4.36 times -1.342 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-\left(-2.139\right)±\sqrt{4.575321+5.85112}}{2\times 1.09}$$

Add 4.575321 to 5.85112 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-\left(-2.139\right)±\sqrt{10.426441}}{2\times 1.09}$$

Take the square root of 10.426441.

$$t=\frac{-\left(-2.139\right)±\frac{3229}{1000}}{2\times 1.09}$$

The opposite of -2.139 is 2.139.

$$t=\frac{2.139±\frac{3229}{1000}}{2\times 1.09}$$

Multiply 2 times 1.09.

$$t=\frac{2.139±\frac{3229}{1000}}{2.18}$$

Now solve the equation t=\frac{2.139±\frac{3229}{1000}}{2.18} when ± is plus. Add 2.139 to \frac{3229}{1000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{\frac{671}{125}}{2.18}$$

Divide \frac{671}{125} by 2.18 by multiplying \frac{671}{125} by the reciprocal of 2.18.

$$t=\frac{1342}{545}$$

Now solve the equation t=\frac{2.139±\frac{3229}{1000}}{2.18} when ± is minus. Subtract \frac{3229}{1000} from 2.139 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$t=-\frac{\frac{109}{100}}{2.18}$$

Divide -\frac{109}{100} by 2.18 by multiplying -\frac{109}{100} by the reciprocal of 2.18.

$$t=-\frac{1}{2}$$

The equation is now solved.

$$t=\frac{1342}{545} t=-\frac{1}{2}$$

Variable t cannot be equal to -\frac{1}{2}.

$$t=\frac{1342}{545}$$

Steps for Completing the Square

Multiply 1.684 and 0.5 to get 0.842.

$$0.842+\frac{1.684\times 0.5}{1.684-1.684\times 0.5}=0.545t+\frac{0.5+t}{1+\frac{4}{2}t}$$

Multiply 1.684 and 0.5 to get 0.842.

$$0.842+\frac{0.842}{1.684-1.684\times 0.5}=0.545t+\frac{0.5+t}{1+\frac{4}{2}t}$$

Multiply 1.684 and 0.5 to get 0.842.

$$0.842+\frac{0.842}{1.684-0.842}=0.545t+\frac{0.5+t}{1+\frac{4}{2}t}$$

Subtract 0.842 from 1.684 to get 0.842.

$$0.842+\frac{0.842}{0.842}=0.545t+\frac{0.5+t}{1+\frac{4}{2}t}$$

Divide 0.842 by 0.842 to get 1.

$$0.842+1=0.545t+\frac{0.5+t}{1+\frac{4}{2}t}$$

Add 0.842 and 1 to get 1.842.

$$1.842=0.545t+\frac{0.5+t}{1+\frac{4}{2}t}$$

Divide 4 by 2 to get 2.

$$1.842=0.545t+\frac{0.5+t}{1+2t}$$

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

$$0.545t+\frac{0.5+t}{1+2t}=1.842$$

Variable t cannot be equal to -\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 2t+1.

$$0.545t\left(2t+1\right)+0.5+t=1.842\left(2t+1\right)$$

Reorder the terms.

$$0.545t\left(2t+1\right)+t+0.5=1.842\left(2t+1\right)$$

Use the distributive property to multiply 0.545t by 2t+1.

$$1.09t^{2}+0.545t+t+0.5=1.842\left(2t+1\right)$$

Combine 0.545t and t to get 1.545t.

$$1.09t^{2}+1.545t+0.5=1.842\left(2t+1\right)$$

Use the distributive property to multiply 1.842 by 2t+1.

$$1.09t^{2}+1.545t+0.5=3.684t+1.842$$

Subtract 3.684t from both sides.

$$1.09t^{2}+1.545t+0.5-3.684t=1.842$$

Combine 1.545t and -3.684t to get -2.139t.

$$1.09t^{2}-2.139t+0.5=1.842$$

Subtract 0.5 from both sides.

$$1.09t^{2}-2.139t=1.842-0.5$$

Subtract 0.5 from 1.842 to get 1.342.

$$1.09t^{2}-2.139t=1.342$$

Divide both sides of the equation by 1.09, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{1.09t^{2}-2.139t}{1.09}=\frac{1.342}{1.09}$$

Dividing by 1.09 undoes the multiplication by 1.09.

$$t^{2}+\left(-\frac{2.139}{1.09}\right)t=\frac{1.342}{1.09}$$

Divide -2.139 by 1.09 by multiplying -2.139 by the reciprocal of 1.09.

$$t^{2}-\frac{2139}{1090}t=\frac{1.342}{1.09}$$

Divide 1.342 by 1.09 by multiplying 1.342 by the reciprocal of 1.09.

$$t^{2}-\frac{2139}{1090}t=\frac{671}{545}$$

Divide -\frac{2139}{1090}, the coefficient of the x term, by 2 to get -\frac{2139}{2180}. Then add the square of -\frac{2139}{2180} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{2139}{1090}t+\left(-\frac{2139}{2180}\right)^{2}=\frac{671}{545}+\left(-\frac{2139}{2180}\right)^{2}$$

Square -\frac{2139}{2180} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{2139}{1090}t+\frac{4575321}{4752400}=\frac{671}{545}+\frac{4575321}{4752400}$$

Add \frac{671}{545} to \frac{4575321}{4752400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{2139}{1090}t+\frac{4575321}{4752400}=\frac{10426441}{4752400}$$

Factor t^{2}-\frac{2139}{1090}t+\frac{4575321}{4752400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{2139}{2180}\right)^{2}=\frac{10426441}{4752400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{2139}{2180}\right)^{2}}=\sqrt{\frac{10426441}{4752400}}$$

Simplify.

$$t-\frac{2139}{2180}=\frac{3229}{2180} t-\frac{2139}{2180}=-\frac{3229}{2180}$$

Add \frac{2139}{2180} to both sides of the equation.

$$t=\frac{1342}{545} t=-\frac{1}{2}$$

Variable t cannot be equal to -\frac{1}{2}.

$$t=\frac{1342}{545}$$