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If
u^(2)-ut=6 and
t^(2)-ut=-2, what is the value of
2(u-t)^(2) ?

If $u^{2}-u t=6$ and $t^{2}-u t=-2$ , what is the value of $2(u-t)^{2}$ ?

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Composition of linear and quadratic functions: find a value

Full solution

Q. If

u^{2}-u t=6

and

t^{2}-u t=-2

, what is the value of

2(u-t)^{2}

?

Given Equations: We are given two equations: $\newline$ $1$ . $u^{2} - ut = 6$ $\newline$ $2$ . $t^{2} - ut = -2$ $\newline$ We need to find the value of $2(u-t)^{2}$ . $\newline$ First, let's express $(u-t)^{2}$ in terms of $u$ and $t$ . $\newline$ $(u-t)^{2} = u^{2} - 2ut + t^{2}$
Express $(u-t)^{2}$ : Now, let's add the two given equations to eliminate the $-ut$ term: $\newline$ $(u^{2} - ut) + (t^{2} - ut) = 6 - 2$ $\newline$ $u^{2} + t^{2} - 2ut = 4$ $\newline$ Notice that $u^{2} + t^{2} - 2ut$ is the expanded form of $(u-t)^{2}$ .
Substitute $(u-t)^{2}$ Term: Since we have found that $u^{2} + t^{2} - 2ut = 4$ , we can say that: $(u-t)^{2} = 4$ Now, we need to find the value of $2(u-t)^{2}$ . $2(u-t)^{2} = 2 \times 4$
Calculate $2(u-t)^{2}$ : Calculate the final value: $\newline$ $2(u-t)^{2} = 2 \times 4 = 8$

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