Quadratic Equation Question :?

Question

Sum of the squares of two nos. is 130. The sum of the smaller and twice the larger no. is 25. Determine the nos.

Answer 1

let x be the smaller number
let y be the larger number

x^2 + y^2 = 130
x + 2y = 25

x = -2y + 25

(-2y + 25)^2 + y^2 = 130

4y^2 - 100y + 625 + y^2 - 130 = 0

5y^2 - 100y + 495 = 0

use quadratic formula and you'll get y = 9 and 11

x^2 + 9^2 = 130
x = 7

x^2 + 11^2 = 130
x = 3

9 and 7 OR 11 and 3

Answer 2

Let a be the first number and b be the second one:

Eq_1) a² + b² = 130
Eq_2) a + 2b = 25

Here you have a simple system of 2 equation and 2 unknowns.

Find "a" in terms of "b" using the second equation: a = -2b + 25

now substitute this equivalant into the first equation:
(-2b + 25)² + b² = 130 simplify:
=> 5b² - 100b + 625 = 130 divide both sides by 5
=> b² - 20b + 125 = 26 simplify:
=> b² - 20b + 99 = 0

Solving this equation yields:
b = 9 & 11

Trying number 9 in one of the equations:
a = 7

Trying number 11 in one of the equations:
a = 3

So: the numbers we are looking for can be (3,11) or (7,9).
Both satisfy our conditions and fit into both equations.

Good Luck!

Answer 3

7 and 9

7 * 7 = 49
9 * 9 = 81
81 + 49 = 130
(9 * 2) + 7 = 25

Answer 4

Let nos. be x and y
x² + y² = 130
x + 2y = 25

x = 25 - 2y
(25 - 2y)² + y² = 130
625 - 100y + 5y² = 130
5y² - 100y + 495 = 0
y² - 20y + 99 = 0
(y - 11).(y - 9) = 0
y = 11, y = 9
x = 3, 7
Numbers are (3,11) and (7,9)

Answer 5

first tell us what nos. is