The sum of the squares of two numbers is 233 and one of the numbers is 3 less than twice the other number. Find the numbers.
Answers
let the first no.be x
other no.is 2x-3
so (2x-3)^2+x^2 =233
4x^2-12x+9+x^2=233
on simplifying and transposing we get
5x^2-12x-224=0
Trying to factor by splitting the middle term
1.1 Factoring 5x2-12x-224
The first term is, 5x2 its coefficient is 5 .
The middle term is, -12x its coefficient is -12 .
The last term, "the constant", is -224
Step-1 : Multiply the coefficient of the first term by the constant 5 • -224 = -1120
Step-2 : Find two factors of -1120 whose sum equals the coefficient of the middle term, which is -12 .
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -40 and 28
5x2 - 40x + 28x - 224
Step-4 : Add up the first 2 terms, pulling out like factors :
5x • (x-8)
Add up the last 2 terms, pulling out common factors :
28 • (x-8)
Step-5 : Add up the four terms of step 4 :
(5x+28) • (x-8)
Which is the desired factorization
x=8/-28/5
2x-3=15(if x=8)or -71/5(if x=-28/5)
Given:The sum of the squares of two numbers is 233.
One of the numbers is 3 less than twice the other number.
To find: the two numbers.
We assume that the numbers are integers and one of the numbers is x.
Thus, the second number will be .
It is given that the sum of the squares of two numbers is 233.
Expanding the equation, we get,
We know that, to determine the roots of any equation, the value of atleast one of the products must be zero.
Thus, we consider the first product of the equation as zero.
Here, the second product is zero.
Thus,
Also, we let x be an integer.
Thus, it call not be a rational number/fraction.
Hence, the value of x is expressed as x = 8
The second number is calculated as,
Hence, the two numbers are 8 and 13.