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Understand real, equal, and complex roots in quadratic equations by analyzing the discriminant through examples.

---

## Understanding Quadratic Equation Roots

In a quadratic equation $$ax^2 + bx + c = 0$$ (where $$a \neq 0$$), the roots of the equation are the values of $$x$$ that make the equation true. A quadratic equation always has two roots that can be found using the formula:

Visible text: In a quadratic equation (where ), the roots of the equation are the values of that make the equation true. A quadratic equation always has two roots that can be found using the formula:

```math
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
```

The value $$b^2 - 4ac$$ is called the determinant or discriminant (denoted by $$D$$), which is very important because it determines the type of roots of the quadratic equation.

Visible text: The value is called the determinant or discriminant (denoted by ), which is very important because it determines the type of roots of the quadratic equation.

### Different Real Roots

If $$D > 0$$ (or $$b^2 > 4ac$$), then the quadratic equation has two different real roots.

Visible text: If (or ), then the quadratic equation has two different real roots.

```math
x_1 = \frac{-b + \sqrt{D}}{2a} \quad \text{and} \quad x_2 = \frac{-b - \sqrt{D}}{2a}
```

Example: $$x^2 - 5x + 6 = 0$$

Visible text: Example:

- $$a = 1, b = -5, c = 6$$
- $$D = (-5)^2 - 4 \cdot 1 \cdot 6 = 25 - 24 = 1 > 0$$
- The roots are: $$x_1 = 3$$ and $$x_2 = 2$$

Visible text: - 
- 
- The roots are: and

### Equal (Repeated) Roots

If $$D = 0$$ (or $$b^2 = 4ac$$), then the quadratic equation has one real repeated root (two roots with the same value).

Visible text: If (or ), then the quadratic equation has one real repeated root (two roots with the same value).

```math
x_1 = x_2 = \frac{-b}{2a}
```

Example: $$x^2 - 6x + 9 = 0$$

Visible text: Example:

- $$a = 1, b = -6, c = 9$$
- $$D = (-6)^2 - 4 \cdot 1 \cdot 9 = 36 - 36 = 0$$
- The roots are: $$x_1 = x_2 = 3$$

Visible text: - 
- 
- The roots are:

### Imaginary (Non-Real) Roots

If $$D < 0$$ (or $$b^2 < 4ac$$), then the quadratic equation has two different complex (imaginary) roots.

Visible text: If (or ), then the quadratic equation has two different complex (imaginary) roots.

```math
x_1 = \frac{-b + i\sqrt{|D|}}{2a} \quad \text{and} \quad x_2 = \frac{-b - i\sqrt{|D|}}{2a}
```

Where $$i = \sqrt{-1}$$ is the imaginary number.

Visible text: Where is the imaginary number.

Example: $$x^2 + 2x + 5 = 0$$

Visible text: Example:

- $$a = 1, b = 2, c = 5$$
- $$D = 2^2 - 4 \cdot 1 \cdot 5 = 4 - 20 = -16 < 0$$
- The roots are: $$x_1 = -1 + 2i$$ and $$x_2 = -1 - 2i$$

Visible text: - 
- 
- The roots are: and

## Relationship Between Roots and Coefficients

If $$x_1$$ and $$x_2$$ are the roots of the quadratic equation $$ax^2 + bx + c = 0$$, then:

Visible text: If and are the roots of the quadratic equation , then:

Component: MathContainer
Children:

```math
x_1 + x_2 = -\frac{b}{a}
```

```math
x_1 \times x_2 = \frac{c}{a}
```

This is an important relationship that can be used to find the coefficients of a quadratic equation if its roots are known.