Question

plz deae solve this problem​

Answer 1

$Given \: \: Question \: \: \: Is \: \\ \\ \alpha \: \: \: and \: \: \: \beta \: \: are \: \: the \: \: \: the \: \: roots \: \: of \: \: the \: \: equation \\ 2x {}^{2} - 5x + 2 = 0 \\ find \: \: ( \alpha - 1) {}^{( \beta - 1)} \\ \\ Answer \: \: \\ \\ 2x {}^{2} - 4x - 1x + 2 = 0 \\ \\ 2x(x - 2) - 1(x - 2) = 0 \\ \\ (x - 2)(2x - 1) = 0 \\ \\ (x - 2) = 0 \: \: \: \: or \: \: \: (2x - 1) = 0 \: \: \: \\ \\ x = 2 \: \: \: \: or \: \: \: \: x = \frac{1}{2} \\ \\ let \: \: \: \alpha = 2 \: \: \: \: and \: \: \: \: \beta = \frac{1}{2} \\ \\ \\ therefore \: \: \: \: ( \alpha - 1) {}^{( \beta - 1)} \: \: = (2 - 1) {}^{( \frac{1}{2} - 1)} \\ \\ ( \alpha - 1) {}^{( \beta - 1)} = 1 {}^{\frac{(1 - 2)}{2} } \\ \\ ( \alpha - 1) {}^{( \beta - 1)} = 1 {}^{ \frac{ - 1}{2} } \\ \\ ( \alpha - 1) {}^{( \beta - 1)} = \frac{1}{1 {}^{ \frac{1}{2} } } \\ \\ ( \alpha - 1) {}^{( \beta - 1)} = 1$

Answer 2

Question: If α and β are the roots of the equation 2x² - 5x + 2 = 0, then

$\sf (\alpha - 1)^\left({\beta - 1}\right)$ = ______ where (α > β).

Solution:

Since the polynomial has been given, we'll try to find the zeroes (α and β) by splitting the middle term.

$\Longrightarrow \sf 2x^2 - 5x + 2 = 0\\ \\ \\\ \sf Sum \dashrightarrow \sf -5 \\ \\ \sf Product \dashrightarrow 4\\ \\\sf Split \dashrightarrow -4 \times -1\\ \\ \\\Longrightarrow \sf 2x^2 - 4x - x + 2 = 0\\ \\\Longrightarrow \sf 2x(x - 2) -1(x - 2) = 0\\ \\\Longrightarrow \sf (2x - 1)(x - 2) = 0$

Taking the zeroes to be 'α' and 'β' we get,

$\Longrightarrow \sf 2x - 1 = 0\\ \\ \Longrightarrow \sf 2x = 1\\ \\ \Longrightarrow \sf x = \frac{1}{2}$

Other root,

$\Longrightarrow \sf x - 2 = 0\\ \\ \Longrightarrow \sf x = 2$

ATQ, α > β, therefore, α = 2, and β = ¹/₂

$\Longrightarrow \sf (\alpha - 1)^{\left({\beta - 1}\right)}\\ \\ \\\Longrightarrow \sf \left(2 - 1\right)^{\left(\dfrac{1}{2} \ - \ 1}\right)}\\ \\ \\\Longrightarrow \sf \left(1\right)^{\left( \dfrac{1 - 2}{2} \right)}\\ \\ \\\Longrightarrow \sf \left(1\right)^{\left(\dfrac{-1 \ \ }{2}\right)}\\ \\ \\\Longrightarrow \sf \ \underline{\textbf{1}}$

∴ The answer is 1.