91Ó°ÊÓ

In mathematics, when we talk about finding real solutions, we're seeking values that satisfy an equation without involving imaginary numbers. Real solutions exist on the number line, unlike complex or imaginary solutions. In many cases, especially when dealing with real-world applications, real solutions are the ones we care about the most.

For example, when finding real solutions to a given system of equations like the one in the exercise, you determine values for variables that satisfy each equation simultaneously.

Consider the example with two equations:
\[\begin{align*}2x^{2} - 5x - 4y^{2} &= -4 \x^{2} - 3y^{2} &= 4\end{align*}\]
The goal is to find pairs \((x, y)\) that make both equations true, leading to real solutions like \((\sqrt{7}, 1)\) or \((-\sqrt{7}, -1)\). Finding real solutions often involves several algebraic steps to simplify equations and determine viable values.

A quadratic equation is a type of polynomial equation of degree two. It generally takes the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(aeq 0\).

Quadratics can appear in a variety of scenarios, like the system of equations in our exercise. By rearranging the second equation to express one variable in terms of another (e.g., \(x^2 = 4 + 3y^2\)), you're essentially converting it into a quadratic format in terms of \(y\):

1. Substitute this expression in the other equation.
2. You might end up with a new quadratic equation to solve, like \(2u^2 - 5u + 12 = 0\).

Solutions to quadratic equations can be found using methods like factoring, completing the square, or applying the quadratic formula. However, not all quadratic equations will provide real solutions. Each solution depends on the discriminant \(b^2 - 4ac\). If the discriminant is positive, you get two distinct real solutions. If zero, one real solution, and if negative, no real solutions.

The substitution method is a key technique to solve systems of equations. It involves expressing one variable in terms of another from one equation, then substituting that expression into the other equation.

Here's how it works:

• Take the given equations, and solve one of them for one variable in terms of the other. For the equation \(x^{2} - 3y^{2} = 4\), you can rewrite it as \(x^{2} = 4 + 3y^{2}\).
  
• Substitute this expression into the other equation. So, \(2x^{2} - 5x - 4y^{2} = -4\) becomes \(2(4 + 3y^{2}) - 5\sqrt{4 + 3y^{2}} - 4y^{2} = -4\).
  
• By simplifying, you form a new equation that may be easier to solve, often reducing the number of variables or complexity.
  

The benefits of the substitution method include its simplicity for smaller systems and its ability to reduce the problem to a single-variable equation. However, it requires careful algebraic manipulation to avoid errors, and when done properly, it aids in finding possible solutions for both variables.