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A quadratic equation is a polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). It is called quadratic because "quad" refers to a square, indicating the highest degree term \( x^2 \). Quadratic equations can be solved by various methods such as factoring, using the quadratic formula, or completing the square.
In this exercise, the first equation after substitution transforms into \( 2x^4 - 18x^2 + 16 = 0 \). Though it doesn't look like a traditional quadratic equation at first glance, it can be treated as one by considering \( x^2 \) as a temporary variable.
This approach simplifies solving by letting us apply quadratic methods. Solving the resulting equation yields the possible values for \( x^2 \), which we then use to find the specific solutions for \( x \). These solutions can then be explored further to find corresponding \( y \)-values in the system of equations.

The substitution method is a popular technique for solving systems of equations, especially when dealing with non-linear equations like quadratic equations. The key idea is to solve one equation for one variable and substitute that expression into the other equation. This reduces the system of equations to a single equation with one unknown.
In this particular system, the substitution method begins by expressing \( y \) from the second equation \( xy = 4 \). This is straightforward, yielding \( y = \frac{4}{x} \).

• First, solve for one variable in one of the equations.
• Second, substitute this expression into the other equation. Here, \( y = 4/x \) is substituted into \( 2x^2 + y^2 = 18 \).
• Finally, solve the resulting equation, which in this case becomes a quadratic equation.

This simplification allows us to find all possible solutions by solving a single equation, making it a very powerful tool in dealing with systems involving quadratic equations.

Algebraic expressions are combinations of numbers, variables, and mathematical operations like addition, multiplication, and exponentiation. Understanding how to manipulate these expressions is fundamental in solving equations.
For example, in our exercise, we began by solving for \( y \) as \( 4/x \). Then, we substituted this expression into the quadratic equation, resulting in \( 2x^4 + 16 = 18x^2 \).

• Algebraically, we simplified this to \( 2x^4 - 18x^2 + 16 = 0 \), using operations like subtraction and multiplication to eliminate fractions.
• This simplifies the complexity of the original system and allows for the application of quadratic solution strategies.

Recognizing and manipulating algebraic expressions is essential. It allows us to reframe and reduce complex problems into more manageable forms that can be solved systematically using known mathematical techniques.