The sum of 2 positive integers is 151. The lesser number is 19 more than the square root of the greater number, What is the value of the?
The sum of two positive integers is 151. The lesser number is 19 more than the square root of the greater number. What is the value of the difference between the greater number and the lesser number?
6 Answers
Jan 20, 2025
x² + x + 19 = 151
x² + x – 132 = 0
x = 11 (valid) or -12 (invalid, as per the question)
The smaller number is 11+19 = 30 and the greater number is 121.
Jan 26, 2025
y = 151 – x
y = x^(1/2) + 19
151 – x = x^(1/2) + 19
x^(1/2) = 132 – x
x = x^2 – 264x + 132^2
x^2 – 265x + 17424 = 0
(x – 144)(x – 121) = 0
Solutions:
x = 144, y = 7
x = 121, y = 30
The numbers are 144 and 7 or 121 and 30
The value of the greater number minus
the lesser number is either 137 or 91.
Feb 14, 2025
A great way to start word problems is to write down everything that is in words in the problem in mathematical terms
– The sum of 2 positive integers is 151
(eq1)
a + b = 151
– The lesser number is 19 more than the square root of the greater number
(eq2)
a – 19 = squareroot(b)
now you can just plug in. Let’s set equation 2 to b= (I tried a= and it was pretty complicated)
b = (a-19)(a-19)
b = a^2 -38a + 361
and plug that into equation 1
a + (a^2 -38a + 361) = 151
set that equation = zero so that we can use the quadratic formula
a^2 – 37a + 210 = 0
I leave the quadratic formula to you. Remember that both a and b must be positive, so you should use the positive value, and not the negative value, that you get from the quadratic formula.
Just plug the value of a into equation 1 to get the value of b.
a + b^2 = 151
a = b + 19
b + 19 + b^2 = 151
b^2 + b – 132 = 0
b = (-1 +/- sqrt(1 + 528)) / 2
b = (-1 +/- sqrt(529)) / 2
b = (-1 +/- 23) / 2
b = -24/2 , 22/2
b = -12 , 11
a and b > 0
b = 11
b^2 = 121
a = 30
30 and 121 are your numbers
You have x+y = 115
and x = 19 + √y or x – 19 = √y.
Therefore x^2 – 38x + 281 = y – 115 – x
Solve the quadratic for x.
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