Equation solve the equation uov(t) u(t)=2t^2-1 v(t)=t-1

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find the new composed function, hence, you need to replace the equation of v(t) for t in equation of u(t), such that:

`(uov)(t) = 2(t - 1)^2 - 1 => (uov)(t) = 2(t^2 - 2t + 1) - 1`

`(uov)(t) = 2t^2 - 4t + 1`

`{((uov)(t) = 0),((uov)(t) = 2t^2 - 4t + 1):} => 2t^2 - 4t + 1 = 0`

You may complete the square `2t^2 - 4t` such that:

`2t^2 - 4t + 2 = -1 + 2 => (sqrt 2*t - sqrt 2)^2 = 1 => sqrt 2*t - sqrt 2 = +-sqrt 1 => sqrt 2*t - sqrt 2 = +-1`

`sqrt 2*t = sqrt 2 +- 1 => t_1 = (sqrt 2 + 1)/sqrt 2`

`t_2 = (sqrt 2 - 1)/sqrt 2 `

Hence, evaluating the solutions to obtained quadratic equation yields `t_1 = (sqrt 2 + 1)/sqrt 2, t_2 = (sqrt 2 - 1)/sqrt 2.`

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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To solve the equation means to determine t, we'll have to find (uov)(t).

(uov)(t) = u(v(t))

We'll substitute t by v(t):

u(v(t)) = 2[v(t)]^2 - 1

u(v(t)) = 2(t-1)^2 - 1

We'll expand the square:

u(v(t)) = 2t^2 - 4t + 2 - 1

We'll combine like terms:

u(v(t)) = 2t^2 - 4t + 1

We'll put u(v(t)) = 0:

2t^2 - 4t + 1 = 0

We'll apply quadratic formula:

t1 = [4+sqrt(16 - 8)]/4

t1 = (4+2sqrt2)/4

t1 = (2+sqrt2)/2

t2 = (2-sqrt2)/2

The values of t for u(v(t)) = 0 are: {(2-sqrt2)/2 ; (2+sqrt2)/2}

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