The distance the bicycle travels and the time taken are expressed by the formula d(t)=t^2-2t, there d(t) is in miles and t in hours. Find the time taken by the bicycle to cover a distance of 63 miles.

just plug in the numbers:

63 = t^2 - 2t
t^2 - 2t - 63 = 0
(t-9)(t+7) = 0

so, t=9 or -7

Since negative time does not correspond to any known reality, use t=9.

Check:
9^2 - 2*9 = 81-18 = 63

Well, to find the time taken by the bicycle to cover a distance of 63 miles, we can set up the equation d(t) = 63 and solve for t.

So, let's substitute d(t) with 63 in the equation:
63 = t^2 - 2t

Now, we have a quadratic equation. To solve it, let's set it to zero by subtracting 63 from both sides:
t^2 - 2t - 63 = 0

Now we have a quadratic equation in standard form. We can solve it by factoring, completing the square, or using the quadratic formula.

Or, we could just ask the bicycle directly, "Hey bicycle, how long did it take you to travel 63 miles?" and hope it can answer us. I've heard some bikes are quite talkative.

To find the time taken by the bicycle to cover a distance of 63 miles, we need to solve the equation d(t) = 63.

Given that d(t) = t^2 - 2t, we can substitute this into the equation:

t^2 - 2t = 63

Rearranging the equation, we get:

t^2 - 2t - 63 = 0

To solve this quadratic equation, we can either factor it or use the quadratic formula.

Factoring the equation, we find:

(t - 9)(t + 7) = 0

Setting each factor equal to zero, we get:

t - 9 = 0 or t + 7 = 0

Solving for t in each equation, we find:

t = 9 or t = -7

Since time cannot be negative, we can discard -7 as a valid solution. Therefore, the time taken by the bicycle to cover a distance of 63 miles is:

t = 9 hours.

To find the time taken by the bicycle to cover a distance of 63 miles, we can set up the equation d(t) = 63 and solve for t.

Given that d(t) = t^2 - 2t, we substitute d(t) with 63 and set up the equation:

63 = t^2 - 2t

To solve this quadratic equation, we rearrange it to be in the form of a quadratic equation and then solve for t.

t^2 - 2t - 63 = 0

Since this equation cannot be easily factored, we can use the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / 2a

For our equation, a = 1, b = -2, and c = -63. Plugging in these values, we get:

t = (-(-2) ± √((-2)^2 - 4(1)(-63))) / (2 * 1)

Simplifying further:

t = (2 ± √(4 + 252)) / 2

t = (2 ± √256) / 2

t = (2 ± 16) / 2

Thus, we have two possible solutions for t:

t1 = (2 + 16) / 2 = 18 / 2 = 9

t2 = (2 - 16) / 2 = -14 / 2 = -7

Since time cannot be negative, we discard the negative solution (-7).

Therefore, the time taken by the bicycle to cover a distance of 63 miles is 9 hours.