Question

If α and β are the zeroes of the polynomial x2−8x+k such that α2+β2=40, find ′k′.​

Answer 1

Answer:

k = 12

Step-by-step explanation:

α and β are the zeroes of the polynomial x²−8x+k

∴ product of the zeroes of the polynomial x²−8x+k is

   αβ = k / 1= k

∴ sum of the zeroes of the polynomial x² - 8x + k is

    α + β = -(-8)/1 = 8

∴ (α + β)² = α² + β² + 2αβ

(8)² = (40) + 2k

64 - 40 = 2k

k = 24/2 = 12

Answer 2

${ \huge{ \underline{ \underline{ \sf{ \green{GivEn : }}}}}}$

• A polynomial __ f(x) = x² - 8x + k

• α² + β² = 40

• α and β are the zeroes of the polynomial.

${ \huge{ \underline{ \underline{ \sf{ \green{To \: find :}}}}}}$

• What's the value of k?

Formula to be used :-

• (a + b) ² = a² + b² + 2ab

• (a + b)² - 2ab = a² + b²

${ \huge{ \underline{ \underline{ \sf{ \green{SoluTion : }}}}}}$

Given polynomial ___

f(x) = x² - 8x + k

Where,

a = 1

b = -8

c = k

We know,

Sum of zeroes = - b/a

Product of zeroes = c/a

_______________________________________________

Hence,

Sum of zeroes = - b/a

⠀⠀⠀⠀⠀⟶ α + β = -(-8)/1

⠀⠀⠀⠀⠀⟶ α + β = 8

product of zeroes = c/a

⠀⠀⠀⠀⠀⟶ αβ = k

_______________________________________________

Now, find the value of k

⟶(α + β)² - 2α β = α² + β²

⠀⠀⠀⠀⠀ ⟶ 8² -2× k = 40

⠀⠀⠀⠀⠀ ⟶ 64 - 2k = 40

⠀⠀⠀⠀ ⠀⟶ - 2k = 40 - 64

⠀⠀⠀⠀⠀ ⟶ - 2k = -24

⟶k = 12

Hence, the value of k is = 12

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