Let x₁, x₂, x₃, x₄ be the solution of the equation 4 x⁴+8 x³-17 x²-12 x+9=0 and (4+x₁²)(4+x₂²)(4+x₃²)(4+x₄²)=125/16 m. Then the value of m is _________.

Solution

Maths sir · 3 steps

Split each 4 + xᵢ² as (xᵢ - 2i)(xᵢ + 2i), and the whole product is just the polynomial at ± 2i.

1. Use the factored form. Write P(x) = 4x⁴ + 8x³ - 17x² - 12x + 9 = 4(x - x₁)(x - x₂)(x - x₃)(x - x₄). Since 4 + xᵢ² = (xᵢ - 2i)(xᵢ + 2i), the product needs only P(2i) and P(-2i). Each of these carries a factor 4, so together they give 16 Π (4 + xᵢ²).

   Πₖ₌₁⁴(4 + xₖ²) = (P(2i) P(-2i))/16 = |P(2i)|²/16

2. Evaluate P at 2i. With x = 2i we get x² = -4, x³ = -8i and x⁴ = 16. So P(2i) = 64 - 64i + 68 - 24i + 9 = 141 - 88i, and |P(2i)|² = 141² + 88² = 19881 + 7744 = 27625.
3. So. The product is 27625/16 = 125/16m, so m = 27625/125 = 221. As a check, the roots are -3, -1, 1/2, 3/2, and 13 × 5 × 17/4 × 25/4 = 27625/16 as well.

The common slip: 3536. Forgetting to divide P(2i)P(-2i) by 4² = 16 for the leading coefficient makes the product 27625, which gives m = 3536.

The idea to keep. A product of the form Π (a² + xᵢ²) over the roots equals P(ai)P(-ai) divided by the leading coefficient squared. You never need the roots themselves.

Answer from the NTA final answer key. Solution by PrepWiser.