Luckiest Guess in the World by me.

Luckiest Guess in the World by me.

Explanation

Difficulty: Hard

Graph

Graph

Since

funciton f of x = (x^2) minus (7 times x) + 10

, substituting (t + 1)

(t + 1)

for

into the function yields f of (t+1) = ((t+1)^2) minus (7 times (t+1)) + 10

f of (t+1) = ((t+1)^2) minus (7 times (t+1)) + 10

, or

f of (t + 1) = ((t^2) + (2 times t) + 1) minus ((7 times t) + 7) +10

, or

f of (t + 1) = (t^2) minus (5 times t) + 4

. Therefore t = 1

t = 1

or

t = 4

.

Since

function f of (t + 1) = 0

, it follows that (t^2) minus (5 times t) + 4 = 0

(t^2) minus (5 times t) + 4 = 0

, or

(t minus 1) times (t minus 4) = 0

. Therefore t = 1

t = 1

or

t = 4

.

Another way to solve the question would be to use a dummy variable

. For example, let k = t + 1

k = t + 1

.

function f of (k) = (k^2) minus (7 times k) + 10 = (k minus 5) times (k minus 2)

. Since

k = t + 1

, it follows that function f of (k) = 0

function f of (k) = 0

. So

(k minus 5) times (k minus 2) = 0

, and therefore, k = 5

k = 5

or

k = 2

. Since

t = k minus 1

,

t = 4

or

t = 1

.

This question asks for one possible value of

. Either

or

(not both) satisfy the question being asked. Choose only one correct answer to enter in the grid.

When there is a range of possible correct answers, your gridded response must lie within the range. For example, consider a problem for which all numbers between

and

, exclusive, are correct answers. For this problem, although 4.0002

4.0002

is within the range (4 less than t less than 5)

(4 less than t less than 5)

, its rounded value 4.00

4.00

is not within the range and therefore would not be considered a correct answer to the problem.

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