Quadratic expression
Understanding Quadratic Expressions
A quadratic expression is a polynomial of degree two. The general form of a quadratic expression in one variable is:
$$ ax^2 + bx + c $$
where ( a ), ( b ), and ( c ) are constants, with ( a \neq 0 ), and ( x ) is the variable.
Key Components of a Quadratic Expression
- Leading Coefficient (( a )): This is the coefficient of the ( x^2 ) term. It determines the direction of the parabola when graphed.
- Linear Coefficient (( b )): This is the coefficient of the ( x ) term.
- Constant Term (( c )): This is the constant term with no variable attached to it.
- Degree: The degree of a quadratic expression is always 2, which means the highest power of ( x ) is 2.
- Discriminant (( D )): This is given by ( D = b^2 - 4ac ) and determines the nature of the roots of the quadratic equation ( ax^2 + bx + c = 0 ).
Table of Differences and Important Points
| Feature | Description |
|---|---|
| Degree | Always 2 for a quadratic expression. |
| Graph | Represents a parabola when plotted on a coordinate plane. |
| Axis of Symmetry | The line ( x = -\frac{b}{2a} ) is the axis of symmetry for the parabola. |
| Vertex | The highest or lowest point on the graph, given by ( \left(-\frac{b}{2a}, -\frac{D}{4a}\right) ). |
| Discriminant | Determines the nature of the roots: real and distinct, real and equal, or complex. |
| Factored Form | Can be expressed as ( a(x - r_1)(x - r_2) ) where ( r_1 ) and ( r_2 ) are the roots of the quadratic equation. |
| Standard Form | The expression is written as ( ax^2 + bx + c ). |
| Vertex Form | The expression can also be written as ( a(x - h)^2 + k ) where ( (h, k) ) is the vertex of the parabola. |
Formulas Related to Quadratic Expressions
- Standard Form: ( ax^2 + bx + c )
- Vertex Form: ( a(x - h)^2 + k )
- Factored Form: ( a(x - r_1)(x - r_2) )
- Axis of Symmetry: ( x = -\frac{b}{2a} )
- Vertex Coordinates: ( \left(-\frac{b}{2a}, -\frac{D}{4a}\right) )
- Discriminant: ( D = b^2 - 4ac )
Examples to Explain Important Points
Example 1: Standard Form and Vertex
Consider the quadratic expression ( 2x^2 - 4x + 1 ).
- Standard Form: It is already in standard form.
Vertex: To find the vertex, we calculate the axis of symmetry and the ( y )-coordinate of the vertex.
- Axis of symmetry: ( x = -\frac{-4}{2 \cdot 2} = 1 )
- ( y )-coordinate of the vertex: ( y = 2(1)^2 - 4(1) + 1 = -1 )
So the vertex is ( (1, -1) ).
Example 2: Discriminant and Nature of Roots
Consider the quadratic expression ( x^2 - 6x + 9 ).
- Discriminant: ( D = (-6)^2 - 4(1)(9) = 36 - 36 = 0 )
- Nature of Roots: Since ( D = 0 ), the quadratic equation ( x^2 - 6x + 9 = 0 ) has real and equal roots.
Example 3: Factored Form
Consider the quadratic expression ( x^2 - 5x + 6 ).
- Factored Form: To factor the expression, we look for two numbers that multiply to ( c ) (which is 6) and add up to ( b ) (which is -5). These numbers are -2 and -3.
So the factored form is ( (x - 2)(x - 3) ).
Example 4: Vertex Form
Consider the quadratic expression ( 3x^2 + 12x + 12 ).
Completing the Square: To write this in vertex form, we complete the square.
- Factor out the leading coefficient from the first two terms: ( 3(x^2 + 4x) + 12 )
- Add and subtract the square of half the coefficient of ( x ): ( 3(x^2 + 4x + 4 - 4) + 12 )
- Simplify: ( 3(x + 2)^2 - 12 + 12 )
- Vertex Form: ( 3(x + 2)^2 )
The vertex is ( (-2, 0) ).
Understanding quadratic expressions is crucial for solving quadratic equations, analyzing graphs of quadratic functions, and for many applications in physics, engineering, economics, and other fields.