Question

Let a, b and c be the roots of the equation x3 – 2x2 + 4x + 10 = 0, then find the value of 1/2(a + 2)(b + 2)(c + 2).

Answer 1

a ,b and c be the roots of equation x³ - 2x² + 4x + 10 = 0.

then, x³ - 2x² + 4x + 10 = (x - a)(x - b)(x - c) = 0

x³ - 2x² + 4x + 10 = (x - c)(x² - (a + b)x + ab)

= x³ - (a + b)x² + abx - cx² + c(a + b)x - abc

= x³ - (a + b + c)x² + (ab + bc + ca)x - abc

now compare both sides,
a + b + c = -2
ab + bc + ca = 4
-abc = 10

we have to find 1/2 (a + 2)(b + 2)(c + 2)

= 1/2 (2 + a)(2 + b)(2 + c)

= 1/2 {2³ + (a + b + c)2² + (ab + bc + ca)2 + abc}

= 1/2 {8 + 4 × -2 + 4 × 2 - 10}

= 1/2 { 8 - 10} = -1

Answer 2

Answer:

7

Step-by-step explanation:

Vieta's theorem states that if a,b,c are the roots of cubic equation

x^3 + px^2 + qx + r = 0 then the roots of this equation is given by

a + b + c = - p

ab + bc + ca = q

abc = - r

Now we have the equation

  x^3 - 2x^2 + 4x + 10 = 0. Comparing the values with the given equation

      p = -2 , q = 4, r = 10

a + b + c = - (- 2) = 2

ab + bc + ca = 4

 abc = - 10

Now we can find the value of

 1/2 (a + 2)(b + 2)(c + 2)

1/2 (ab + 2b + 2a + 4)(c + 2)

1/2 [abc + 2(ab + bc + ca) + 4(a + b + c) + 8]

1/2[-10 + 2(4) + 4(2) + 8]

1/2[-10 + 8 + 8 + 8]

1/2(14) = 7

Thus the value is 7