Question

The sum of the squares of two consecutive natural numbers is 25. Represent
this situation in the form of a quadratic equation.
a) x2 + (x+1)2 = 25
b) x2 - (x+1)=25
c) (x+12 - x2 = 25
d) x2+ (x+1)2 - 25 = 0
​

Answer 1

✬ Equation : x² + (x + 1)² = 25 ✬

Given:

• Sum of squares of two consecutive natural numbers is 25.

To Show:

• In form of quadratic equation.

Solution: Let first consecutive natural number be x. Therefore,

➟ Second consecutive number = (x + 1)

Now,

• Square of first number = (x)²
• Square of 2nd number = (x + 1)²

A/q

• Sum is 25.

➯ Equation = (x)² + (x + 1)² = 25

Hence option A is correct.

_______________________

$\implies{\rm }$ x² + (x + 1)² = 25

$\implies{\rm }$ x² + (x² + 1² + 2•x•1) = 25

$\implies{\rm }$ x² + x² + 1 + 2x = 25

$\implies{\rm }$ 2x² + 2x = 25 – 1

$\implies{\rm }$ 2x² + 2x = 24

$\implies{\rm }$ 2x² + 2x – 24 = 0

$\implies{\rm }$ 2(x² + x – 12)

$\implies{\rm }$ x² + x – 12

Now, break this by Middle term splitting

➙ x² + x – 12

➙ x² + 4x – 3x – 12

➙ x(x + 4) – 3 (x + 4)

➙ (x – 3) (x + 4)

➙ x – 3 = 0 or, x + 4 = 0

➙ x = 3 or x = –4

We will take positive value of x. { Negative ignored }

So, The two consecutive natural numbers are

➮ First number = x = 3

➮ Second number = x + 1 = 3 + 1 = 4

Answer 2

$\huge{\mathbb{\red{ANSWER}}}$

Equation : x² + (x + 1)² = 25 ✬

$\huge{\mathbb{\red{Given}}}$

Sum of squares of two consecutive natural numbers is 25.

$\huge{\mathbb{\red{To find}}}$

In form of quadratic equation.

Solution: Let first consecutive natural number be x. Therefore,

➟ Second consecutive number = (x + 1)

Now,

Square of first number = (x)²

Square of 2nd number = (x + 1)²

A/q

Sum is 25.

➯ Equation = (x)² + (x + 1)² = 25

Hence option A is correct.

_______________________

\implies{\rm }⟹ x² + (x + 1)² = 25

\implies{\rm }⟹ x² + (x² + 1² + 2•x•1) = 25

\implies{\rm }⟹ x² + x² + 1 + 2x = 25

\implies{\rm }⟹ 2x² + 2x = 25 – 1

\implies{\rm }⟹ 2x² + 2x = 24

\implies{\rm }⟹ 2x² + 2x – 24 = 0

\implies{\rm }⟹ 2(x² + x – 12)

\implies{\rm }⟹ x² + x – 12

Now, break this by Middle term splitting

➙ x² + x – 12

➙ x² + 4x – 3x – 12

➙ x(x + 4) – 3 (x + 4)

➙ (x – 3) (x + 4)

➙ x – 3 = 0 or, x + 4 = 0

➙ x = 3 or x = –4

We will take positive value of x. { Negative ignored }

So, The two consecutive natural numbers are

➮ First number = x = 3

➮ Second number = x + 1 = 3 + 1 = 4