Sure, I'll give a theoretical explanation of a general mathematical problem. Let's consider a classic problem from algebra: solving a quadratic equation.

Theoretical Explanation: Solving a Quadratic Equation

A quadratic equation is a second-order polynomial equation in a single variable $x$ with the general form: $ax^2 + bx + c = 0$ where $a$, $b$, and $c$ are constants with $a \neq 0$.

1. Standard Form

The standard form of a quadratic equation is $ax^2 + bx + c = 0$.

2. Solutions of the Quadratic Equation

The solutions to the quadratic equation, also known as the roots, can be found using several methods:

a. Factoring: If the quadratic equation can be factored, it can be written as: $(mx + n)(px + q) = 0$ where $m$, $n$, $p$, and $q$ are constants. Setting each factor to zero gives the solutions: $mx + n = 0 \implies x = -\frac{n}{m}$ $px + q = 0 \implies x = -\frac{q}{p}$

b. Completing the Square: This method involves rewriting the quadratic equation in the form of a perfect square trinomial. The steps are as follows:

1. Divide all terms by $a$ (if $a \neq 1$).
2. Move the constant term to the right side of the equation.
3. Add and subtract $(\frac{b}{2a})^2$ to the left side.
4. Write the left side as a squared binomial.
5. Solve for $x$.

Example: $x^2 + 6x + 8 = 0$ Move constant: $x^2 + 6x = -8$ Add and subtract $(\frac{6}{2})^2 = 9$: $x^2 + 6x + 9 = 1$ Rewrite as a binomial square: $(x + 3)^2 = 1$ Solve for $x$: $x + 3 = \pm1 \implies x = -2, -4$

c. Quadratic Formula: The quadratic formula provides a direct way to find the solutions of any quadratic equation: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ This formula is derived from the process of completing the square and works for all quadratic equations.

d. Graphical Method: The quadratic equation can be represented graphically as a parabola. The solutions are the points where the parabola intersects the $x$-axis.

3. Discriminant

The discriminant of the quadratic equation $ax^2 + bx + c = 0$ is given by: $D = b^2 - 4ac$ The discriminant tells us about the nature of the roots:

• If $D > 0$, there are two distinct real roots.
• If $D = 0$, there is one real root (or a repeated real root).
• If $D < 0$, there are two complex roots.

Example Problem:

Solve the quadratic equation $2x^2 - 4x + 2 = 0$ using the quadratic formula.

Solution: Here, $a = 2$, $b = -4$, and $c = 2$.

Calculate the discriminant: $D = (-4)^2 - 4(2)(2) = 16 - 16 = 0$

Since $D = 0$, there is one real root. Using the quadratic formula: $x = \frac{-(-4) \pm \sqrt{0}}{2(2)} = \frac{4 \pm 0}{4} = 1$

The solution is $x = 1$.

Further Study Questions:

1. What are the properties of the graph of a quadratic function?
2. How does changing the coefficients $a$, $b$, and $c$ affect the graph of the quadratic function?
3. How can the quadratic formula be derived from completing the square?
4. What are some real-world applications of quadratic equations?
5. How do you solve a quadratic equation using the method of completing the square?
6. How do you solve a quadratic equation by factoring?
7. What is the significance of the vertex of a parabola?
8. How do complex roots of a quadratic equation appear graphically?

Tip: When solving quadratic equations, always start by checking if the equation can be factored easily. Factoring can often provide a quicker solution than the quadratic formula or completing the square.