Question

solve the quadratic equations : the difference between two natural number is 4 the sum of their squares is 170. find the numbers.​

Answer 1

Given

x-y=4 ------->1

(x^2)+(y^2)=170

As we know:-

(a-b)^2=(a^2)-2ab+(b^2)

From this,

(x-y)^2=(x^2)-2xy+(y^2)

(4^2)=(x^2)-2xy+(y^2)

(4^2)=(x^2)+(y^2)-2xy

16=170-2xy

2xy=170-16

2xy=154

xy=154/2

xy=77

Now we know:-

(a+b)^2=(a^2)+2ab+(b^2)

From this,

(x+y)^2=(x^2)+2xy+(y^2)

(x+y)^2=(x^2)+(y^2)+2xy

(x+y)^2=170+(2×77)

(x+y)^2=170+154

(x+y)^2=324

(x+y)=√324

x+y=18 -------->2

From all this two questions we got:-

x-y=4

x+y=18

Add both of this:-

x-y+x+y=4+18

2x=22

x=22/2

x=11

As we know,

x-y=4

11-y=4

y=11-4

y=7

Therefore x=11 and y=7.

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Answer 2

Answer:

• 7 and 11

Step-by-step explanation:

Given

• Two natural numbers differ by 4
• Sum of their squares is 170

To find

• The numbers

Solution

Let the numbers be (x) and (x + 4)

ATP,

• (x)² + (x + 4)² = 170
• 2x² + 8x -154 = 0
• x² + 4x - 77 = 0
• x(x + 11) - 7(x + 11) = 0
• (x + 11) (x - 7) = 0
• x = -11 or x = 7

(x + 4)² :-

• (a + b)²
• a² + b² + 2ab
• x² + 4² + 2(x)(4)
• x² + 16 + 8x
• x² + x² + 16 + 8x
• 2x² + 16 + 8x = 170
• 2x² + 8x - 154 = 0

x cannot be negative, since x is 7.

Hence, the required numbers are 7 and 11.