Question
quadratic equations
Answers
GPT-4o mini
A quadratic equation is a polynomial equation of the form:
\[ ax^2 + bx + c = 0 \]
where:
- \( a \), \( b \), and \( c \) are constants (with \( a \neq 0 \)),
- \( x \) represents the variable or unknown.
### Key Features of Quadratic Equations
1. **Degree**: The degree of a quadratic equation is 2, as the highest exponent of the variable \( x \) is 2.
2. **Graph**: The graph of a quadratic equation is called a parabola. It opens upwards if \( a > 0 \) and downwards if \( a < 0 \).
3. **Roots**: The solutions to the equation \( ax^2 + bx + c = 0 \) are called the roots of the quadratic equation. They can be found using various methods:
- **Factoring**: If the quadratic can be factored easily, it can be expressed as:
\[ (px + q)(rx + s) = 0 \]
Then, set each factor to zero to solve for \( x \).
- **Completing the Square**: This involves rearranging the equation to form a perfect square.
- **Quadratic Formula**: The roots can be found using the quadratic formula:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]
where the term \( b^2 - 4ac \) is called the **discriminant**. It determines the nature of the roots:
- If \( D > 0 \): Two distinct real roots.
- If \( D = 0 \): One real root (a repeated root).
- If \( D < 0 \): No real roots (two complex roots).
### Examples
1. Solve the quadratic equation:
\[ 2x^2 - 4x - 6 = 0 \]
**Using the Quadratic Formula:**
\[ a = 2, b = -4, c = -6 \]
\[ D = (-4)^2 - 4(2)(-6) = 16 + 48 = 64 \]
\[ x = \frac{{4 \pm \sqrt{64}}}{{4}} = \frac{{4 \pm 8}}{4} \]
This gives:
\[ x = 3 \quad \text{and} \quad x = -1 \]
2. Factor the quadratic equation:
\[ x^2 - 5x + 6 = 0 \]
**Factoring**:
\[ (x - 2)(x - 3) = 0 \]
Hence, the roots are:
\[ x = 2 \quad \text{and} \quad x = 3 \]
### Applications
Quadratic equations are used in various fields, including physics, engineering, finance, and statistics. They can model a variety of contexts, such as projectile motion, profit maximization, and various optimization problems. If you have specific scenarios or types of problems regarding quadratic equations, feel free to ask!
\[ ax^2 + bx + c = 0 \]
where:
- \( a \), \( b \), and \( c \) are constants (with \( a \neq 0 \)),
- \( x \) represents the variable or unknown.
### Key Features of Quadratic Equations
1. **Degree**: The degree of a quadratic equation is 2, as the highest exponent of the variable \( x \) is 2.
2. **Graph**: The graph of a quadratic equation is called a parabola. It opens upwards if \( a > 0 \) and downwards if \( a < 0 \).
3. **Roots**: The solutions to the equation \( ax^2 + bx + c = 0 \) are called the roots of the quadratic equation. They can be found using various methods:
- **Factoring**: If the quadratic can be factored easily, it can be expressed as:
\[ (px + q)(rx + s) = 0 \]
Then, set each factor to zero to solve for \( x \).
- **Completing the Square**: This involves rearranging the equation to form a perfect square.
- **Quadratic Formula**: The roots can be found using the quadratic formula:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]
where the term \( b^2 - 4ac \) is called the **discriminant**. It determines the nature of the roots:
- If \( D > 0 \): Two distinct real roots.
- If \( D = 0 \): One real root (a repeated root).
- If \( D < 0 \): No real roots (two complex roots).
### Examples
1. Solve the quadratic equation:
\[ 2x^2 - 4x - 6 = 0 \]
**Using the Quadratic Formula:**
\[ a = 2, b = -4, c = -6 \]
\[ D = (-4)^2 - 4(2)(-6) = 16 + 48 = 64 \]
\[ x = \frac{{4 \pm \sqrt{64}}}{{4}} = \frac{{4 \pm 8}}{4} \]
This gives:
\[ x = 3 \quad \text{and} \quad x = -1 \]
2. Factor the quadratic equation:
\[ x^2 - 5x + 6 = 0 \]
**Factoring**:
\[ (x - 2)(x - 3) = 0 \]
Hence, the roots are:
\[ x = 2 \quad \text{and} \quad x = 3 \]
### Applications
Quadratic equations are used in various fields, including physics, engineering, finance, and statistics. They can model a variety of contexts, such as projectile motion, profit maximization, and various optimization problems. If you have specific scenarios or types of problems regarding quadratic equations, feel free to ask!