Question

Sum of the squares of two consecutive odd numbers is 394.
a) If the first number is , write down the second number in terms of .
b) Write an equation including using the given information.

Answer 1

Answer:

Hint: Consider the 2 consecutive odd numbers as x and x+2. Find the sum of squares of these numbers.Find the value of x and you will get the two consecutive odd numbers.

“Complete step-by-step answer:”

The sum of the squares of 2 consecutive odd numbers is 394. Let us consider one odd number as x and the other consecutive odd number as (x + 2).

We know the odd numbers 1, 3, 5, 7……

So if one number is ‘x’ then the other consecutive odd number can be found by adding 2 to the 1st

1st

number.

So let us take 2 consecutive odd numbers as x and x + 2.

Now it is given that the sum of squares of these consecutive numbers x and (x + 2) is 394.

∴(x)2+(x+2)2=394

∴(x)2+(x+2)2=394

We know, (a+b)2=a2+2ab+b2

(a+b)2=a2+2ab+b2

.

Now open the brackets and simplify them,

x2+x2+2×2x+22=394⇒2x2+4x+4=394

x2+x2+2×2x+22=394⇒2x2+4x+4=394

Divide the entire equation by 2.

x2+2x+2=197x2+2x=197−2x2+2x=195x2+2x−195=0−(1)

x2+2x+2=197x2+2x=197−2x2+2x=195x2+2x−195=0−(1)

We got a quadratic equation which is similar to the general quadratic equation, ax2+bx+c=0

ax2+bx+c=0

.

By comparing equation (1) and the general equation, we get

a = 1, b = 2, c = -195.

Apply these values in the quadratic formula −b±b2−4ac−−−−−−−√2a

−b±b2−4ac2a

and find the value of x.

−2±(2)2−4×1×(−195)−−−−−−−−−−−−−−−−−√2×1=−2±4+780−−−−−−√2=−2±784−−−√2=−2±28×28−−−−−−√2=−2±282

−2±(2)2−4×1×(−195)2×1=−2±4+7802=−2±7842=−2±28×282=−2±282

Hence the roots are (−2+282)

(−2+282)

and (−2−282)

(−2−282)

= 13 and -15.

∴

∴

Value of x = 13, which is an odd number.

Thus we got the 1st

1st

consecutive number as x =13.

Hence, 2nd

2nd

consecutive number as x + 2 = 13 + 2 = 15

Thus the 2 consecutive odd numbers are 13 and 15.

Note: You should consider 2 consecutive terms as x and (x + 2), which is the key to solve this question. We know an odd number, for example 3 is an odd number. (3 + 2) gives 5, which is the odd number near to 3. Thus, 3 and 5 are consecutive terms.

Answer 2

Answer:

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Step-by-step explanation: