# Quadratic System

Pro Problems > Math > Algebra > Equations > Systems of Equations > Non-Linear## Quadratic System

Find all ordered pairs (x, y) such that the following two equations are true:

x^{2} - 4y^{2} = 108

x = 18 - 2y

## Solution

In order to make it feasible for teachers to use these problems in their classwork, no solutions are publicly visible, so students cannot simply look up the answers. If you would like to view the solutions to these problems, you must have a Virtual Classroom subscription.## Similar Problems

### Product of X and Y

For the ordered pair (x,y) the product of x and y is 108. If x + 2y = 30, find all possible ordered pairs (x,y).

### X and Y Quadratics

Find the sum of x and y, if the following are true:

(x + 2)(x - 1) = (y - 12)(y + 3)

(x + 1)(x + 3) = (y - 5)(y - 7)

### Sum of X and Y

Find the sum of x and y if x and y are positive numbers such that x^{2} + 3xy + y^{2} = 424 and xy = 100

### Cubic and Linear

Find all ordered pairs (x,y) which solve the following system of equations:

x^{3} + 12xy^{2} = 7x^{2}y

x + y = 20

### System with Radical

The sum of two numbers is seven times the difference between three times the second number and twice the first number. If the second number is subtracted from the first, the result is the square root of the first. Find all possible values for the first number.

### Linear and Quadratic

Find all ordered pairs (x, y) such that:

3x - y = 10

x^{2} + 8x - y^{2} + 3y = 17

### To Sum It Up

I have picked three positive integers for the lottery, as follows: The sum of my numbers is 54. The sum of my numbers, plus the sum of two of my numbers, is 84. The sum of the squares of my numbers is 1034. What are the three integers?

### System with a Product

Find all ordered pairs (x, y) such that

2x + xy + y = 18

x - y = 2

### Sum and Product System

The sum of a number and twice another number is ten less than the product of the numbers. The sum of the numbers is ten. What are all possible numbers that satisfy these criteria?

### One Equation, Two Variables

Usually we say that if we have two variables, we need two equations to solve, and if we have three variables, we need three equations to solve. This is not 100% true, however, and the problem below is a good example of a single equation in two variables which produce a single ordered pair solution.

Solve for the ordered pair (x,y) such that x^{2} + y^{2} - 2x + 4y = -5