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QuestionSolve the equation t+1+1t=0t + 1 + \frac{1}{t} = 0 and express solutions as a+bia + bi.

Studdy Solution

STEP 1

Assumptions1. The equation is t+t+1=0t^ + t +1 =0 . The solutions are complex numbers of the form a+bia + bi, where aa and bb are real numbers and ii is the imaginary unit with the property that i=−1i^ = -1.

STEP 2

The given equation is a quadratic equation. We can solve it using the quadratic formula, which is given byt=−b±b2−4ac2at = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}where aa, bb, and cc are the coefficients of the quadratic equation at2+bt+c=0at^2 + bt + c =0.

STEP 3

In our equation t2+t+1=0t^2 + t +1 =0, we have a=1a =1, b=1b =1, and c=1c =1. Plugging these values into the quadratic formula, we gett=−1±12−⋅1⋅12⋅1t = \frac{-1 \pm \sqrt{1^2 - \cdot1 \cdot1}}{2 \cdot1}

STEP 4

implify the expression under the square root.
t=−1±1−42t = \frac{-1 \pm \sqrt{1 -4}}{2}

STEP 5

Calculate the value under the square root.
t=−1±−32t = \frac{-1 \pm \sqrt{-3}}{2}

STEP 6

The square root of a negative number is imaginary. We can express −3\sqrt{-3} as 3i\sqrt{3}i, where ii is the imaginary unit. Thus, we havet=−1±3i2t = \frac{-1 \pm \sqrt{3}i}{2}

STEP 7

Finally, we can express the solutions in the form a+bia + bi by dividing the real and imaginary parts by2 separately.
t=−12±32it = -\frac{1}{2} \pm \frac{\sqrt{3}}{2}iThe solutions of the equation are t=−12+32it = -\frac{1}{2} + \frac{\sqrt{3}}{2}i and t=−12−32it = -\frac{1}{2} - \frac{\sqrt{3}}{2}i.

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