Formation of QE
Formation of Quadratic Equations (QE)
Quadratic equations are polynomial equations of degree two. The general form of a quadratic equation is:
$$ ax^2 + bx + c = 0 $$
where ( a ), ( b ), and ( c ) are constants, and ( a \neq 0 ).
Understanding the Components
- ( a ): This is the coefficient of ( x^2 ) and determines the concavity of the parabola that represents the equation graphically. If ( a > 0 ), the parabola opens upwards, and if ( a < 0 ), it opens downwards.
- ( b ): This is the coefficient of ( x ) and affects the location of the vertex of the parabola along the x-axis.
- ( c ): This is the constant term and affects the location of the parabola along the y-axis.
Formation of a Quadratic Equation
A quadratic equation can be formed given different sets of information:
- Given the roots: If the roots of the equation are ( \alpha ) and ( \beta ), the quadratic equation can be written as:
$$ (x - \alpha)(x - \beta) = 0 $$
Expanding this, we get:
$$ x^2 - (\alpha + \beta)x + \alpha\beta = 0 $$
- Given a point and the vertex: If the vertex of the parabola is ( (h, k) ) and it passes through the point ( (x_1, y_1) ), the equation can be formed using the vertex form:
$$ a(x - h)^2 + k = y $$
Solving for ( a ) using the point ( (x_1, y_1) ), we can find the specific equation.
- Given three points: If the parabola passes through three points ( (x_1, y_1) ), ( (x_2, y_2) ), and ( (x_3, y_3) ), we can set up a system of equations to solve for ( a ), ( b ), and ( c ).
Differences and Important Points
| Aspect | Description |
|---|---|
| Degree | A quadratic equation always has a degree of 2. |
| Number of Roots | It can have two real roots, one real root (repeated), or two complex roots. |
| Discriminant | Given by ( D = b^2 - 4ac ), it determines the nature of the roots. |
| Sum and Product of Roots | For roots ( \alpha ) and ( \beta ), ( \alpha + \beta = -\frac{b}{a} ) and ( \alpha\beta = \frac{c}{a} ). |
| Graphical Representation | The graph of a quadratic equation is a parabola. |
Formulas
- Standard Form: ( ax^2 + bx + c = 0 )
- Factored Form: ( a(x - \alpha)(x - \beta) = 0 )
- Vertex Form: ( a(x - h)^2 + k = y )
- Discriminant: ( D = b^2 - 4ac )
Examples
Example 1: Formation from Roots
Given the roots ( \alpha = 3 ) and ( \beta = -2 ), form the quadratic equation.
$$ (x - \alpha)(x - \beta) = (x - 3)(x + 2) = 0 $$ $$ x^2 + 2x - 3x - 6 = 0 $$ $$ x^2 - x - 6 = 0 $$
Example 2: Formation from a Point and Vertex
Given the vertex ( (2, 3) ) and a point ( (4, 11) ), form the quadratic equation.
Using the vertex form:
$$ a(x - h)^2 + k = y $$ $$ a(x - 2)^2 + 3 = y $$
Substitute the point ( (4, 11) ):
$$ a(4 - 2)^2 + 3 = 11 $$ $$ 4a + 3 = 11 $$ $$ a = 2 $$
The quadratic equation is:
$$ 2(x - 2)^2 + 3 = y $$ $$ 2x^2 - 8x + 8 + 3 = y $$ $$ 2x^2 - 8x + 11 = y $$
Example 3: Formation from Three Points
Given points ( (1, 2) ), ( (2, 3) ), and ( (3, 10) ), form the quadratic equation.
Set up the system of equations:
$$ a(1)^2 + b(1) + c = 2 $$ $$ a(2)^2 + b(2) + c = 3 $$ $$ a(3)^2 + b(3) + c = 10 $$
Solving this system, we find the values of ( a ), ( b ), and ( c ) that satisfy all three equations, resulting in the specific quadratic equation.
Understanding the formation of quadratic equations is crucial for solving various mathematical problems, including those involving optimization, projectile motion, and geometry. The ability to form and manipulate these equations is a foundational skill in algebra.