Question

please help me how to solve this?​

Answer 1

Given : (1 + m)x² - 2(1 + 3m)x + (1 + 8m) has equal roots

★  A Quadratic Equation ax² + bx + c = 0 has equal roots only if discriminant is zero, where discriminant is given by b² - 4ac

$\bigstar\;\;\textsf{Condition for an Equation to have Equal roots is : \boxed{\mathsf{b^2 - 4ac = 0}}}$

Consider the Equation : (1 + m)x² - 2(1 + 3m)x + (1 + 8m) = 0

here : a = (1 + m) and b = -2(1 + 3m) and c = (1 + 8m)

$:\implies$  [-2(1 + 3m)]² - 4(1 + m)(1 + 8m) = 0

$:\implies$  4(1 + 3m)² - 4(1 + 8m + m + 8m²) = 0

$:\implies$  4(1 + 9m² + 6m) - 4(1 + 9m + 8m²) = 0

$:\implies$  4 + 36m² + 24m - 4 - 36m - 32m² = 0

$:\implies$  4m² - 12m = 0

$:\implies$  4m(m - 3) = 0

$:\implies$  4m = 0  (or)  m - 3 = 0

$:\implies$  m = 0  (or)  m = 3

Given : α and β are values of m of (1 + m)x² - 2(1 + 3m)x + (1 + 8m)

★  α = 0  $:\implies$  α + 2 = (2 + 0) = 2

★  β = 3  $:\implies$  β + 2 = (3 + 2) = 5

If Δ and Ф are roots of a Quadratic Equation then the equation can be represented as :

★  x² - (Δ + Ф)x + ΔФ = 0

Quadratic Equation whose roots are (α + 2) and (β + 2) is :

$:\implies$  x² - (2 + 5)x + (2)(5) = 0

$:\implies$  x² - 7x + 10 = 0