91Ó°ÊÓ

Quadratic equations are a type of polynomial equation where the highest exponent of the variable is 2. They are in the standard form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants.
Solving quadratic equations often involves finding the values of \(x\) which satisfy the equation.
The quadratic equation present in our exercise is \(y = x^2\). This equation describes a parabola that opens upward when graphed in the coordinate plane. The vertex of this parabola is at the origin (0, 0), and it is symmetric around the y-axis.
There are various methods to solve quadratic equations, including:

• Factoring
• Completing the square
• Using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

In our case, we simplified the problem by setting the quadratic equation equal to the reciprocal function.

A reciprocal function is a function of the form \(y = \frac{1}{x}\). This type of function has some unique properties:

• The graph of \(y = \frac{1}{x}\) is a hyperbola with two asymptotes: the x-axis (y = 0) and the y-axis (x = 0).
• The function is undefined at \(x = 0\) since division by zero is not possible.
• The function is symmetric with respect to the origin, meaning it has rotational symmetry of 180° about the origin.

In the context of our exercise, we used \(y = \frac{1}{x}\) and equated it to the quadratic function \(y = x^2\).

This allowed us to combine them into a single equation that we could solve using algebraic manipulation.

Algebraic manipulation involves the use of various techniques to rearrange and simplify equations to solve for a particular variable.
In the exercise, we started with the system of equations:

• y = x^2
• y = \frac{1}{x}\

By setting these two equations equal to each other, we obtained:

\(x^2 = \frac{1}{x}\)

To eliminate the fraction, we multiplied both sides by \(x\):

\(x^3 = 1\).

We then solved for \(x\) by finding the cube root, yielding \(x = 1\) and \(x = -1\).

Finally, we substituted these values of \(x\) back into the original equations to find the corresponding \(y\) values and verified each solution in both equations for consistency.

Algebraic manipulation is a powerful tool in solving various types of equations and is essential in many areas of mathematics. It includes operations like:

• Combining like terms
• Using the distributive property
• Factoring and expanding expressions

These steps help in simplifying and solving complex equations step by step.