Question

if alpha and beta are zeroes of the quadratic polynomial 2x² -3x +4 then find the value of alpha/ beta + beta /alpha​

Answer 1

EXPLANATION.

α,β are the zeroes of quadratic polynomial,

⇒ p(x) = 2x² - 3x + 4.

Sum of zeroes of quadratic polynomial.

⇒ α + β = -b/a.

⇒ α + β = -(-3)/2 = 3/2. .....(1).

Products of zeroes of quadratic polynomial.

⇒ αβ = c/a.

⇒ αβ = 4/2 = 2. .....(2).

To find ⇒ α/β + β/α.

Take L.C.M in this equation, we get.

⇒ α² + β²/αβ.

Formula of (a² + b²) = (a + b)² - 2ab.

⇒ (α + β)² - 2αβ/αβ.

Put the values in this equation, we get.

⇒ (3/2)² - 2(2)/2.

⇒ (9/4) - 4/2.

⇒ 9 - 16/4/2.

⇒ -7/4/2.

⇒ -7/4 X 1/2.

⇒ -7/8.

Value of (α/β + β/α) = -7/8.

                                                                                                 

MORE INFORMATION.

Conditions for common roots,

Let quadratic equations are,

(1) = a₁x² + b₁x + c₁ = 0.

(2) = a₂x² + b₂x + c₂ = 0.

(1) = If only one roots is common,

⇒ x = b₁c₂ - b₂c₁ / a₁b₂ - a₂b₁.

⇒ y = c₁a₂ - c₂a₁ / a₁b₂ - a₂b₁.

(2) = If both roots are common,

⇒ a₁/a₂ = b₁/b₂ = c₁/c₂.

                                                                                                   

Answer 2

$\large\underline{\red{\sf \orange{\bigstar} Correct\; Question}}$

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• if alpha and beta are zeroes of the quadratic polynomial 2x² -3x +4 then find the value of alpha/ beta + beta /alpha

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$\large\underline{\blue{\sf \pink{\bigstar} Solution}}$

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☯ Given expression

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ➪ p(x) = 2x² - 3x + 4.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\pink{\sf{\star\;⚛ \;Sum\; of\; zeroes\; of \;quadratic\; polynomial.}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ➪ α + β = -b/a.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ➪ α + β = -(-3)/2 = 3/2. .....(1).

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\red{\sf{\star\;⚛\; Products \;of\; zeroes \;of \;Quadratic\; Polynomial}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ➪ αβ = c/a.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ➪ αβ = 4/2 = 2. .....(2).

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• To find ⇒ α/β + β/α.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ➪ α² + β²/αβ.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\blue{\sf{\star\;☯\; Formula \;of\; (a² + b²)\; = \;(a + b)²\; - 2ab.}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ➪ (α + β)² - 2αβ/αβ.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\purple{\sf{\star\;⚛ \;Put\; the\; values\; in\; this\; equation,\; we\; get.}}$

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• ➪(3/2)² - 2(2)/2.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ➪ (9/4) - 4/2.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ➪ 9 - 16/4/2.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ➪ -7/4/2.

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• ➪-7/4 X 1/2.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• ➪ -7/8.${\boxed{\green{\checkmark{}}}}$

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Value of (α/β + β/α) = -7/8.${\boxed{\pink{\checkmark{}}}}$

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