91Ó°ÊÓ

Quadratic equations are a type of polynomial equation where the highest exponent of the variable is 2. They typically take the form \(ax^2 + bx + c = 0\). These equations are significant in mathematics because they can model a wide range of phenomena, from projectile motion to optimization problems.
In the context of the exercise, we come across a quadratic equation when we substitute \(x = y - 2\) into the equation \(3x^2 - 10y = 5\). After completing the substitution and simplifying, the equation is transformed into a quadratic in terms of \(y\):
\[3y^2 - 22y + 7 = 0\]
An essential feature of quadratic equations is that they can have either two, one, or no real solutions, depending on the discriminant \(\Delta = b^2 - 4ac\):

• If \(\Delta > 0\), there are two distinct real solutions.
• If \(\Delta = 0\), there is exactly one real solution.
• If \(\Delta < 0\), there are no real solutions.

For our equation, the discriminant helps us determine how many real solutions \(y\) has, guiding us to find the corresponding \(x\) values.

The substitution method is a straightforward approach for solving systems of equations, especially when one of the equations can be easily solved for one of the variables. It involves replacing one variable with an equivalent expression from another equation.
To tackle the problem in the exercise, we start with the system:

• \(3x^2 - 10y = 5\)
• \(x - y = -2\)

By expressing \(x\) in terms of \(y\) from the second equation, \(x = y - 2\), we find a direct relation between the two variables. We then substitute this expression for \(x\) into the first equation:
\[3(y-2)^2 - 10y = 5\]
This technique simplifies the original system to a single equation with one variable, \(y\), which can be solved using methods specific to quadratic equations. Once values for \(y\) are found, they can be substituted back into the expression for \(x\) to find the corresponding \(x\) values.

In mathematical terms, a real solution is a solution to an equation that is a real number, as opposed to a complex or imaginary number. Determining whether a system of equations has real solutions often involves evaluating the discriminant of a quadratic equation.
For a quadratic equation \(ax^2 + bx + c = 0\), its discriminant \(\Delta = b^2 - 4ac\) reveals the nature of the roots:

• \(\Delta > 0\): Two distinct real solutions.
• \(\Delta = 0\): One real solution (also called a repeated or double root).
• \(\Delta < 0\): No real solutions (the solutions are complex or imaginary).

In this exercise, we calculated the discriminant of the quadratic \(3y^2 - 22y + 7 = 0\) and found it to be positive. Thus, the equation has two real solutions for \(y\). These solutions are then used to find corresponding \(x\) values, giving us two solution pairs \((x_1, y_1)\) and \((x_2, y_2)\) for the system of equations. It is crucial to verify that these solutions satisfy both original equations, confirming their validity.