Question

SOLVE THIS CORRECT ANSWER WILL BE MARKED AS BRAINLIEST​

Answer 1

Gɪᴠᴇɴ :-

• x = 1/√[9 + 4√5]

Tᴏ Fɪɴᴅ :-

• x³ + 7x² + 16x + 7 ?

Sᴏʟᴜᴛɪᴏɴ :-

Lets First Try to Solve Denominator Inside Square Root Part :-

→ 9 + 4√5

→ 5 + 4 + 2 * 2 * √5

→ (√5)² + (2)² + 2 * 2 * √5

Comparing it with a² + b² + 2ab now,

→ (√5 + 2)²

So,

→ x = 1/√[(√5 + 2)²]

→ x = 1/(√5 + 2)

Rationalizing it,

→ x = 1/(√5 + 2) * [ (√5 - 2)/(√5 - 2) ]

→ x = (√5 - 2)/[ (√5)² - (2)² ]

→ x = (√5 - 2) / (5 - 4)

→ x = (√5 - 2)

Now,

→ (x + 2) = √5 ----------- Equation (1)

Squaring both sides we get,

→ (x + 2)² = (√5)²

→ x² + 4x + 4 = 5

→ x² + 4x - 1 = 0 -------- Equation (2)

________________

Now, Splitting The finding part :-

→ x³ + 7x² + 16x + 7

→ x³ + (4x² + 3x²) + (12x + 5x - x) + (10 - 3)

Re - arranging Them now,

→ (x³ + 4x² - x) + (3x² + 12x - 3) + (5x + 10)

→ x(x² + 4x - 1) + 3(x² + 4x - 1) + 5(x + 2)

Putting value of Equation (1) & (2) Now, we get,

→ x * 0 + 3 * 0 + 5 * √5

→ 0 + 0 + 5√5

→ 5√5 (Ans.)

[ Excellent Question. ]

Answer 2

Solution:

Given ,

x=1/{√9+4√5}

x= 1/{√4+5+2.√5.2}

x=1/{√(√5+2)²

x=1/(√5+2)

Now Rationalize,

=> x = 1/(√5+2) × (√5-2)/(√5+2)

=> x = √5-2/√5-4

=> x= √5-2

=> x+ 2=√5

by squaring b/s we get,

=> (x+2)²=(√5)²

=>x²+4x+4=5

=>x²+4x+4-5=0

=> x²+4x-1=0

So,

=> x³+7x²+16x+7

=> x³+4x³+3x²+12x-3+5x-x+10

=>x(x²+4x-1)+3(x²+4x-1)+5(x+2)

=>x(0)+3(0)+5(√5)

=>0+0+5√5

=>5√5