Question

The sum of the Squares of three consecutive number is 110.
Then determine the Three Numbers.​

Answer 1

$\large\bold{{Question :-}}$The sum of the Squares of three consecutive number is 110. Then determine the Three Numbers.

$\large\bold\red{\underline{Answer :-}}$

So, First of all The Question is to find out three consecutive Numbers .

First of all let's learn a little about Consecutive Numbers. So, Consecutive Numbers are the Numbers which follows each other. For example :- 1,2,3,4 are consecutive Numbers.

Let's start with Our Question :-

• Let The First consecutive number be = x
• and second number be = x + 1
• Third number be = x + 2

So,

$:\pink\implies$ $\bold{{(x)^{2}}}$ + $\bold{{(x + 1)^{2}}}$ + $\bold{{(x + 2)^{2}}}$ = $\bold{{110}}$

$:\pink\implies$ $\bold{{(x)^{2}}}$ + $\bold{{(x)^{2}}}$ + $\bold{{(1)^{2}}}$+ $\bold{{2x}}$ +$\bold{{(x)^{2}}}$ + $\bold{{(4)}}$+ $\bold{{(4x)}}$ = $\bold{{110}}$

$:\pink\implies$ $\bold{{3x^{2}}}$ + $\large\bold{{6x}}$ + $\bold{{5}}$ - $\large\bold{{110}}$ = $\bold{{0}}$

$:\pink\implies$ $\bold{{3x^{2}}}$ + $\bold{{6x}}$ - $\bold{{105}}$ = $\bold{{0}}$

By taking 3 Out as Common, we are left with the Quadratic Equation :—

$:\pink\implies$ $\large\bold{{1x^{2}}}$+ $\large\bold{{2x}}$ - $\large\bold{{35}}$ = $\large\bold{{0}}$

By Using Quadratic Formula,

x = -b ± √(b)^2— 4 × a × c / 2a

x = -2 ± √(2)^2— 4 × 1 × -35 / 2×1

x = -2 ± √144/2

[ As we know that "144" is square of 12 ]

x = -2 ± 12/2

• Taking Positive
• x = -2+12/2
• x = 5

Hence, x = +5

x +1 = 5+1 = +6

x+2 = 5+2 = +7

• Taking Negative sign
• x = -2-12/2
• x = -7

Hence, x = -7

x+1 = -7+1 = -6

x+2 = -7+2 = -5

Hence, The Three consecutive Numbers are ±5,±6,±7.

Answer 2

Answer:

• ± (5, 6, 7)

Step-by-step explanation:

Let the three consecutive numbers be denoted as :

1. x
2. x + 1
3. x + 2

Identity

• (x + a)² = x² + 2ax + a²

It is given that the sum of the squares of these numbers add up to 110.

• (x)² + (x + 1)² + (x + 2)² = 110
• x² + x² + 2x + 1 + x² + 4x + 4 = 110
• 3x² + 6x + 5 = 110
• 3x² + 6x - 105 = 0
• x² + 2x - 35 = 0
• x² + 7x - 5x - 35 = 0
• x(x + 7) - 5(x + 7) = 0
• (x - 5)(x + 7) = 0
• x = 5 or x = -7

Taking x = 5

• x² = 5² = 25
• (x + 1)² = 6² = 36
• (x + 2)² = 7² = 49
• ⇒ 25 + 36 + 49 = 110

Taking x = -7

• x² = (-7)² = 49
• (x + 1)² = (-6)² = 36
• (x + 1)² = (-5)² = 25
• ⇒ 49 + 36 + 25 = 110

Solution

• 5, 6, 7 ⇔ -7, -6, -5
• ⇒ ± (5, 6, 7)