Question

if α and β are zeroes of (2x²-5x+3) then find the value of
(a) α/β + β/α
(b) 1/2 alpha + 1/2beta​

Answer 1

Answer:

Value of :

(a) 4/3

(b) 5/4

Step-by-step explanation:

Given equation is:

2x² - 5x + 3

Now,

→ 2x² - 5x + 3

→ 2x² - (2x + 3x) + 3

→ 2x² - 2x - 3x + 3

→ 2x(x - 1) - 3(x - 1)

→ (x - 1)(2x - 3)

Zero's :

• x - 1 = 0

→ x = 1

• 2x - 3 = 0

→ 2x = 3

→ x = 3/2

α and β are 1 and 3/2.

(a) α/β + β/α

→ 1/(3/2) + (3/2)/1

→ 2/3 + 2/3

→ 4/3

(b) 1/2 α + 1/2 β

→ 1/2 × 1 + 1/2 × 3/2

→ 1/2 + 3/4

→ (2 + 3)/4

→ 5/4

Answer 2

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★ Step by step explanation :

• Here, we have a quadratic equation (2x² - 5x + 3), α and β are it's zeroes. We had to find out the value of (a) α/β + β/α , (b) 1/2 α + 1/2 β.

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★ Using middle splitting method :

$\qquad\longrightarrow\:\tt 2x^2 - 5x + 3= 0$

$\qquad\longrightarrow\:\tt 2x^2 - 2x - 3x + 3 = 0$

$\qquad\longrightarrow\:\tt 2x(x - 1) - 3(x - 1) = 0$

$\qquad\longrightarrow\:\tt (x - 1) (2x - 3) = 0$

$\qquad\longrightarrow\:\tt x - 1 = 0 \:,\:2x - 3 = 0$

$\qquad\longrightarrow\:\tt x = \red{1}\:,\:2x = 3$

$\qquad\longrightarrow\:\bf{ x = 1 \:,\:x = \red{\dfrac{3}{2}}}$

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$\therefore\:{\small{\underline{\frak{Zeroes\:of\:the\:polynomial(\alpha\:and\:\beta)\:=\:1\:and\:\frac{3}{2}}}}}$

Now,

★ Solving for (a) and (b) :

• (a) α/β + β/α

$\qquad\dashrightarrow\:\tt \dfrac{1}{\frac{3}{2}} + \dfrac{\frac{3}{2}}{1}$

$\qquad\dashrightarrow\:\tt \dfrac{2}{3} + \dfrac{2}{3}$

$\qquad\dashrightarrow\:\tt \dfrac{2 + 2}{3}$

$\qquad\dashrightarrow\:\bf \red{\dfrac{4}{3}}$

• (b) 1/2 α + 1/2 β

$\qquad\dashrightarrow\:\tt \bigg[\bigg(\dfrac{1}{2}\:\times\:1\bigg) + \bigg(\dfrac{1}{2}\:\times\:\dfrac{3}{2}\bigg)\bigg]$

$\qquad\dashrightarrow\:\tt \dfrac{1}{2} + \dfrac{3}{4}$

$\qquad\dashrightarrow\:\tt \dfrac{2 + 3}{4}$

$\qquad\dashrightarrow\:\bf\red{\dfrac{5}{4}}$

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$\therefore\:{\small{\underline{\frak{Value\:of\:(a)\:\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\:and\:(b)\:\dfrac{1}{2}\alpha + \dfrac{1}{2}\beta\:=\:\frac{4}{3}\:and\:\frac{5}{4}}}}}$

★ More to know :

• α + β = -b/a
• αβ = c/a

Where,

• b is the coefficient of x
• c is constant term
• a is coefficient of x²

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