A quadratic equation solver finds the roots (solutions) of a quadratic equation $$ax^2 + bx + c = 0$$ using methods like factoring, completing the square, or the quadratic formula.

### 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
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 $$x = 1$$

$x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$

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:
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 $$x = 3$$.

#### 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 $$a < 0$$.