Hi. Please i need help with these . No jokes . Question b : Find ;
the values of
1.alpha³+beta³ and 1/(alpha³) +1/(beta³)
2. the the equation whose roots are alpha³ and beta ³

9514 1404 393

  (a) 2, 6

  (b) 155/8, -155, 8x² -155x -1 = 0

Step-by-step explanation:

(a) The equation in standard form is ...

  x^2 -(4+k)x +4k +1 = 0

Then the discriminant is ...

  (4+k)^2 -4(1)(4k+1) = 4^2 +8k +k^2 -16k -4 = k^2 -8k +12 = (k -2)(k -6)

The roots are equal when the discriminant is zero. The values of k that make the discriminant be zero are k=2 and k=6.

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(b) a^3 + b^3 = (a +b)(a^2 -ab +b^2) = (a+b)((a+b)^2 -3ab)

  = (5/2)((5/2)^2 -3(-1/2)) = (5/2)(25/4 +3/2) = 155/8 = α³ +β³

  1/a^3 +1/b^3 = (a^3 +b^3)/(ab)^3 = (155/8)/(-1/2)^3 = -155 = 1/α³ +1/β³

The desired equation can be written from ...

  (x -a^3)(x -b^3) = 0

  x^2 -(a^3 +b^3)x +(ab)^3 = 0

  x^2 -155/8x -1/8 = 0 . . . . .substituting known values

  8x^2 -155x -1 = 0

∠1≅∠2 by the alternate exterior angles theorem.

step-by-step explanation:

given, a ∥ b and ∠1 ≅ ∠3 .we have to prove that e ∥ f

we know that ∠1≅∠3 and that a || b because they are given. we see that by the alternate exterior angles theorem. therefore, ∠2≅∠3 by the transitive property. so, we can conclude that e || f by the converse alternate exterior angles theorem.

we have to fill the missing statement.

transitivity property states that if a = b and b = c, then a = c.

now, given ∠1≅∠3 and by transitivity property ∠2≅∠3 .

hence, to apply transitivity property one angle must be common which is not in result after applying this property which is ∠1.

the only options in which ∠1 is present are ∠1 and ∠2, ∠1 and ∠4

∠1 and ∠4 is not possible ∵ after applying transitivity we didn't get ∠4.

hence, the missing statement is ∠1≅∠2.

so, ∠1≅∠2 by the alternate exterior angles theorem.