Question

The sum of squares of two numbers is 80. The square of the smaller number is 2 times the larger
number, find the two numbers.​

Answer 1

Answer:

Two numbers are:

• Smaller number = ± 4
• Larger number = 8

Step-by-step explanation:

Given that:

• The sum of squares of two numbers is 80.
• The square of the smaller number is 2 times the larger number.

To Find:

• The two numbers.

Let us assume:

• Smaller number be x.
• Larger number be y.

According to the question.

Square of the smaller number = 2 times the larger number

⟶ x² = 2y

⟶ y = x²/2 _____(i)

Sum of squares of two numbers = 80

⟶ x² + y² = 80

Substituting the value of y.

⟶ x² + (x²/2)² = 80

⟶ x² + x⁴/4 = 80

Taking 4 common in LHS.

⟶ (4x² + x⁴)/4 = 80

Cross multiplication.

⟶ 4x² + x⁴ = 80 × 4

⟶ 4x² + x⁴ = 320

⟶ x⁴ + 4x² - 320 = 0

⟶ (x²)² + 4x² - 320 = 0

⟶ (x²)² + 20x² - 16x² - 320 = 0

⟶ x²(x² + 20) - 16(x² + 20) = 0

⟶ (x² - 16) (x² + 20) = 0

⟶ x² = 16 or x² = - 20 (complex number)

⟶ x = √16

⟶ x = ± 4

In equation (i).

When x = 4

⟶ y = x²/2

⟶ y = (4)²/2

⟶ y = 16/2

⟶ y = 8

When x = - 4

⟶ y = x²/2

⟶ y = (- 4)²/2

⟶ y = 16/2

⟶ y = 8

We get that:

• Smaller number = ± 4
• Larger number = 8

Answer 2

Answer:

Given :-

• The sum of squares of two numbers is 80.
• The square of the smaller number is 2 times the larger number.

To Find :-

• What is the number.

Solution :-

Let,

• Larger number be x
• Smaller number be y

According to the question :

➦ Sum of two numbers = 80

↦ x² + y² = 80 - - - - - (Equation No 1)

Now, the square of smaller number = 2 times the larger number.

↦ y² = 2x

↦ x = y²/2 - - - - - (Equation No 2)

Now, by putting the value of x in the equation no 1 we get,

↦ (y²/2)² + y² = 80

↦ y⁴/4 + y² = 80

↦ y⁴ + 4y²/4 = 80

↦ y⁴ + 4y² = 80 × 4

↦ y⁴ + 4y² = 320

↦ y⁴ + 4y² - 320 = 0

↦ y⁴ + (20 - 16)y² - 320 = 0

↦ y⁴ + 20y² - 16y² - 320 = 0

↦ y²(y² + 20) - 16(y² + 20) = 0

↦ (y² + 20)(y² - 16) = 0

↦ y² + 20 = 0

↦ y² = - 20

➠ y² = - 20

Either,

↦ y² - 16 = 0

↦ y² = 16

↦ y = √16

➠ y = 4

Now, by putting the value of y in the equation no 1 we get,

↦ x² + y² = 80

↦ x² + (4)² = 80

↦ x² + 16 = 80

↦ x² = 80 - 16

↦ x² = 64

↦ x = √64

➠ x = 8

∴ The two numbers are 8 and 4.