Question

If the sum of two numbers is 8 and the sum of their square is 40.Find these numbers​

Answer 1

Answer:

The numbers are 2 and 6.

Step-by-step explanation:

Let the numbers be a and b,

Now according to the Question,

Sum of terms = 8

a + b = 8

So,

a = 8 - b ------- 1

Again, according to the Question,

Sum of square of the numbers = 40

a² + b² = 40 -------- 2

Now, If we substitute eq.1 in eq.2 we get,

(8 - b)² + b² = 40

Using the identity,

(x - y)² = x² + 2xy + y²

So,

(8² - 2 × 8 × b + b²) + b² = 40

(64 - 16b + b²) + b² = 40

b² - 16b + 64 + b² = 40

2b² - 16b + 64 = 40

Taking 2 as common factor,

2(b² - 8b + 32) = 40

b² - 8b + 32 = 40/2

b² - 8b + 32 = 20

b² - 8b + 32 - 20 = 0

b² - 8b + 12 = 0

xb² + yb + z = 0

Using Splitting the Middle term method,

Sum = y = (-8)

Product = x × z = 12

So, Factors are (-2) and (-6).

Then,

b² - 2b - 6b + 12 = 0

b(b - 2) - 6(b - 2) = 0

(b - 2)(b - 6) = 0

So,

b = 2 or b = 6

Here, an important step is to find the actual value of b, that is, is it 2 or 6, or is it both.

Now, from eq.1 we get,

a = 8 - b

Let b = 2

Then,

a = 8 - 2 = 6

Then,

a² + b² = 6² + 2²

= 36 + 4

= 40

Let b = 6

Then,

a = 8 - 6 = 2

Thus,

a² + b² = 2² + 6²

= 4 + 36

= 40

Hence,

a = 2 or 6

b = 6 or 2

So, we can say that,

The numbers are 2 and 6.

Hope it helped you and believing you understood it...All the best

Answer 2

Let the numbers be a and b,

Now accordingly,

Sum of terms = 8

a + b = 8

So,

a = 8 - b ------- (1)

Again, according to the Question,

Sum of square of the numbers = 40

a² + b² = 40 -------- (2)

Now, If we substitute eq.1 in eq.2 we get,

(8 - b)² + b² = 40

Using the identity,

(x - y)² = x² + 2xy + y²

So,

(8² - 2 × 8 × b + b²) + b² = 40

(64 - 16b + b²) + b² = 40

b² - 16b + 64 + b² = 40

2b² - 16b + 64 = 40

Taking 2 as common factor,

2(b² - 8b + 32) = 40

b² - 8b + 32 = 40/2

b² - 8b + 32 = 20

b² - 8b + 32 - 20 = 0

b² - 8b + 12 = 0

xb² + yb + z = 0

Now,

Using Splitting the Middle term method,

Sum = y = (-8)

Product = x × z = 12

So, Factors are (-2) and (-6).

Then,

b² - 2b - 6b + 12 = 0

b(b - 2) - 6(b - 2) = 0

(b - 2)(b - 6) = 0

So,

b = 2 or b = 6

Here, to find the actual value of b, that is, is it 2 or 6, or is it both.

Now, from eq.1 we get,

a = 8 - b

Let b = 2

Then,

a = 8 - 2 = 6

So,

a² + b² = 6² + 2²

= 36 + 4

= 40

Let b = 6

Then,

a = 8 - 6 = 2

Thus,

a² + b² = 2² + 6²

= 4 + 36

= 40

Hence,

a = 2,b = 6

So,

The numbers are 2 and 6.