Question

${x}^{2} + 18x + 30 = 2 \sqrt{ {x}^{2} + 18x + 45}$
Find the product of real roots​

Answer 1

GIVEN :–

$\\ \rm \implies{x}^{2} + 18x + 30 = 2 \sqrt{ {x}^{2} + 18x + 45}$

TO FIND :–

• Product of real roots = ?

SOLUTION :–

$\\ \rm \implies{x}^{2} + 18x + 30 = 2 \sqrt{ {x}^{2} + 18x + 45}$

$\\ \rm \implies{x}^{2} + 18x + 30 = 2 \sqrt{ {x}^{2} + 18x + 30 + 15}$

• Let's put x² + 18x + 30 = t –

$\\ \rm \implies t= 2 \sqrt{t+15}$

• Sequare on both sides –

$\\ \rm \implies t^2 = 4(t+15)$

$\\ \rm \implies t^2 = 4t+60$

$\\ \rm \implies t^2 - 4t - 60 = 0$

$\\ \rm \implies t^2 - 10t + 6t- 60 = 0$

$\\ \rm \implies t(t-10)+6(t-10)= 0$

$\\ \rm \implies (t + 6)(t-10)= 0$

$\\ \rm \implies t=10(\checkmark), - 6(x)$

• When t = 10 :–

$\\ \rm \implies {x}^{2} + 18x + 30=10$

$\\ \rm \implies {x}^{2} + 18x + 20 = 0$

• It's has real roots , So –

$\\ \rm \implies Product \: \: of \: \: roots = \dfrac{20}{1} = 20$

• Hence –

$\\\implies \large \pink{\boxed{\rm Product\:\:of\:\:Roots=20}}$

Answer 2

Answer:

Given:-

${x}^{2} + 18x + 30 = 2 \sqrt{ {x}^{2} + 18x + 45}$

To find:-

Product of real roots.

Solution:-

Let,

x² + 18x + 30 = t

$t = 2 \sqrt{t + 15}$

t² - 4t - 60 = 0

$=> t = 10, - 6 ( t \geqslant 0)$

t = 10 = x² + 18x + 30

x² + 18x + 20 = 0

•°• Products of real roots = 20