Question

${\boxed {\LARGE{\mathfrak {solve\:it:-}}}}$

$\tt \sqrt {2x+2\sqrt {2x+4}}=4$​

Answer 1

EXPLANATION.

⇒ √2x + 2√2x + 4 = 4.

As we know that,

Squaring on both sides of the equation, we get.

⇒ (√2x + 2√2x + 4)² = (4)².

⇒ 2x + 2√2x + 4 = 16.

Taking 2x in R.H.S sides of equation, we get.

⇒ 2√2x + 4 = (16 - 2x)

Again squaring on both sides of the equation, we get.

⇒ (2√2x + 4)² = (16 - 2x)².

⇒ 4(2x + 4) = (16 - 2x)².

As we know that,

Formula of :

⇒ (x - y)² = x² + y² - 2xy.

⇒ 8x + 16 = 256 + 4x² - 64x.

⇒ 4x² - 64x + 256 - 8x - 16 = 0.

⇒ 4x² - 72x + 240 = 0.

Taking 4 as common in equation, we get.

⇒ 4(x² - 18x + 60) = 0.

⇒ x² - 18x + 60 = 0.

⇒ D = Discriminant  Or b² - 4ac.

⇒ D = (-18)² - 4(1)(60).

⇒ D = 324 - 240.

⇒ D = 84.

⇒ α = - b + √D/2a.

⇒ α = -(-18) + √84/2.

⇒ α = 18 + √84/2.

⇒ α = 18 + 9.1/2.

⇒ α = 27.1/2.

⇒ α = 13.55.

⇒ β = -b - √D/2a.

⇒ β = -(-18) - √84/2.

⇒ β = 18 - 9.1/2.

⇒ β = 8.9/2.

⇒ β = 4.45.

                                                                                                                         

MORE INFORMATION.

Nature of the factors of the quadratic expression.

(1) = Real and different, if b² - 4ac > 0.

(2) = Rational and different, if b² - 4ac is a perfect square.

(3) = Real and equal, if b² - 4ac = 0.

(4) = If D < 0 Roots are imaginary and unequal or complex conjugate.

Answer 2

$\tt\sqrt {2x+2\sqrt {2x+4}}=4 \\ \\ \implies \rm 2x + 2 \sqrt{2x + 4} = 16 \\ \\ \implies \rm 2 \sqrt{2x + 4} = 16 - 2x \\ \\ \implies \rm2( \sqrt{2x + 4} ) = 2(8 - x) \\ \\ \implies \rm \sqrt{2x + 4} = 8 - x \\ \\ \implies \rm2x + 4 = 64 + {x}^{2} - 16x \\ \\ \implies \rm \: \: \: \: \: \: \: \: \: \: \: 0 = {x}^{2} - 18x + 60 \\ \\ \implies \rm {x}^{2} - 18x + 60 = 0 \\ \tiny{ \text{comparing \: with \: } \rm{ax}^{2} + bx + c = 0 }$

a = 1

b = - 18

c = 60

Now,

$\implies \rm x = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a} \: \: \: \: \: \: \: \: \: \tiny \{ using \: quadratic \: formula \} \\ \\ \implies \rm x = \frac{ - ( - 18) \pm \sqrt{ {( - 18)}^{2} - 4(1)(60) } }{2(1)} \\ \\ \implies \rm x = \frac{18 \pm \sqrt{324 - 240} }{2} \\ \\ \implies \rm x = \frac{18 \pm \sqrt{84} }{2} \\ \\ \implies \rm x = \frac{2(9 \pm \sqrt{21}) }{2} \\ \\ \implies \sf x = \red{ 9 \pm \sqrt{21} }$