Chapter 21: Simultaneous Quadratic Equations

In Chapter 16, we solved systems of linear equations. In this chapter, we expand those techniques to solve systems where at least one equation is a quadratic. These systems typically result in more than one pair of values for x and y.

21.1 One Linear and One Quadratic Equation

The most common case is a system consisting of one linear equation and one quadratic equation. The most effective method here is Substitution.

Procedure:
1. From the linear equation, express one variable in terms of the other.
2. Substitute this expression into the quadratic equation.
3. Solve the resulting quadratic equation for the first variable.
4. Substitute these values back into the linear equation to find the corresponding values of the second variable.

Example: Solve the system:
(1) x + y = 7
(2) x² + y² = 25

Step 1: From (1), y = 7 - x.

Step 2: Substitute into (2):
x² + (7 - x)² = 25
x² + 49 - 14x + x² = 25
2x² - 14x + 24 = 0
x² - 7x + 12 = 0

Step 3: Factor: (x - 3)(x - 4) = 0. So, x = 3 or x = 4.

Step 4: Find y:
If x = 3, y = 7 - 3 = 4.
If x = 4, y = 7 - 4 = 3.

Solutions: (3, 4) and (4, 3)

21.2 Homogeneous Equations

A "Homogeneous" system is one where every term containing a variable in both equations is of the second degree (e.g., terms like x², xy, or y²). These can often be solved by substituting y = vx.

Doctor's Note: When solving simultaneous quadratics, always check both pairs of answers in both original equations. It is easy to find "extraneous" solutions that don't actually work in the context of the original problem.