Quadratic Equation Solver

Introduction

A quadratic equation is a second-degree polynomial equation in a single variable x with the general form:\[ ax^2 + bx + c = 0 \]
where \( a, b, \) and \( c \) are constants and \( a \neq 0 \). The term \( ax^2 \) is called the quadratic term, \( bx \) is the linear term, and \( c \) is the constant term.

Methods of Solving Quadratic Equations

There are several methods to solve quadratic equations:

1. Factoring
2. Completing the Square
3. Quadratic Formula
4. Graphical Method

1. Factoring

This method involves expressing the quadratic equation in the form:
\[ (px + q)(rx + s) = 0 \]
To solve:
1. Factor the quadratic equation into two binomials.
2. Set each factor equal to zero.
3. Solve for x.

Example:

Solve \( x^2 – 5x + 6 = 0 \).

1. Factor: \( (x – 2)(x – 3) = 0 \)
2. Set each factor to zero: \( x – 2 = 0 \) or \( x – 3 = 0 \)
3. Solve: \( x = 2 \) or \( x = 3 \)

2. Completing the Square

This method involves rewriting the quadratic equation in the form:
\[ (x – p)^2 = q \]

To solve:

1. Move the constant term to the other side: \( ax^2 + bx = -c \)
2. Divide by ‘a’ if a ≠ 1.
3. Add and subtract \(\left(\frac{b}{2a}\right)^2\) inside the equation to complete the square.
4. Rewrite as a perfect square trinomial.
5. Solve for x by taking the square root of both sides.

Example:

Solve \( x^2 – 6x + 5 = 0 \).

1. Move the constant: \( x^2 – 6x = -5 \)
2. Add \((\frac{-6}{2})^2 = 9\): \( x^2 – 6x + 9 = 4 \)
3. Rewrite: \( (x – 3)^2 = 4 \)
4. Solve: \( x – 3 = \pm 2 \) → \( x = 5 \) or \( x = 1 \)

3. Quadratic Formula

The quadratic formula provides the solutions to any quadratic equation:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]

To use this method:
1. Identify \( a, b, \) and \( c \) from the equation \( ax^2 + bx + c = 0 \).
2. Substitute into the quadratic formula.
3. Simplify to find the values of \( x \).

Example:

Solve \( 2x^2 + 4x – 6 = 0 \).

1. Identify: \( a = 2, b = 4, c = -6 \)
2. Substitute: \( x = \frac{-4 \pm \sqrt{4^2 – 4 \cdot 2 \cdot (-6)}}{2 \cdot 2} \)
3. Simplify: \( x = \frac{-4 \pm \sqrt{16 + 48}}{4} = \frac{-4 \pm \sqrt{64}}{4} = \frac{-4 \pm 8}{4} \)
4. Solutions: \( x = 1 \) or \( x = -3 \)

4. Graphical Method

This method involves graphing the quadratic function \( y = ax^2 + bx + c \) and finding the x-intercepts (where \( y = 0 \)).

To use this method:
1. Graph the quadratic function.
2. Identify the points where the graph intersects the x-axis.
3. These x-intercepts are the solutions to the quadratic equation.

Example:
Graph \( y = x^2 – 4x + 3 \).

– The x-intercepts are at \( x = 1 \) and \( x = 3 \).

Additional Information

Discriminant

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \Delta = b^2 – 4ac \). It indicates the nature of the roots:

– If \( \Delta > 0 \), the equation has two distinct real roots.
– If \( \Delta = 0 \), the equation has one real root (repeated root).
– If \( \Delta < 0 \), the equation has two complex roots.

Vertex Form

A quadratic equation can also be written in vertex form:
\[ y = a(x – h)^2 + k \]
where \( (h, k) \) is the vertex of the parabola. This form is useful for graphing and understanding the transformation of the parabola.

Parabola

The graph of a quadratic function is a parabola. Important features include:
– Vertex: The highest or lowest point.
– Axis of Symmetry: A vertical line through the vertex.
– Direction: Opens upwards if \( a > 0 \) and downwards if \( a < 0 \).