Question

: the sum of two irrational numbers multiplied by the larger one is 70 and their difference is multiplied by the smaller one is 12

Answer 1

$\huge\sf{Answer}$

Given -

• The sum of two irrational numbers multiplied by a larger one is 70.

• Their difference is multiplied by the smaller one is 12.

To Find -

• The two numbers.

The numbers as -  x and y.

∴ x > y

We have,

⇒ (x + y)(x) = 70 or x² + xy = 70    ......Eq (1)  

⇒ (x - y)(y) = 12 or xy - y² = 12   ......Eq (2)

Adding Eq 1 & 2 -

⇒ x² + y² = 58     .....Eq (3)

⇒ y =√(58 - x²)

Now,

Subsituting the value of above expression in Eq 1 -

⇒ x² + x √(58 - x²) = 70

⇒ x √(58 - x²) = 70 - x²

Taking Squares on both sides,

⇒ x² (58 - x²) = 4900 - 140 x² + x⁴

⇒ 58 x² - x⁴ = 4900 - 140 x² + x⁴

⇒ 4900 - 140 x² + x⁴ = 58 x² - x⁴

⇒ 2x⁴ - 198x² + 4900 = 0

⇒ x⁴ - 99x² + 2450 = 0

⇒ x² = [99 ± √(99² - 4 × 1 × 2450)]/2

⇒ x² = (99 ± 1)/2

⇒ x² = 50, 49

⇒ x = 5 √(2)

⇒ x = 7

From Equation 3 -

⇒ y² = 58 - x² = 58 - 50 = 8

⇒ y = 2 √(2)

∴ The numbers are 5 √(2), 2 √(2)

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