Sign of QE
Understanding the Sign of Quadratic Equations (QE)
Quadratic equations are a fundamental part of algebra and are characterized by the general form:
$$ ax^2 + bx + c = 0 $$
where ( a ), ( b ), and ( c ) are constants, and ( a \neq 0 ). The solutions to a quadratic equation are the values of ( x ) that satisfy the equation, and these solutions are also known as the roots of the equation.
Determining the Sign of Quadratic Equations
The sign of a quadratic equation at a particular value of ( x ) depends on the sign of the quadratic expression ( ax^2 + bx + c ). To understand the sign of a quadratic equation, we need to consider the following factors:
- The sign of the leading coefficient ( a )
- The discriminant ( \Delta ), given by ( \Delta = b^2 - 4ac )
- The roots of the equation, if they are real
Table: Factors Influencing the Sign of a Quadratic Equation
| Factor | Description | Influence on Sign |
|---|---|---|
| Leading Coefficient ( a ) | Determines the direction of the parabola | If ( a > 0 ), parabola opens upwards; if ( a < 0 ), parabola opens downwards |
| Discriminant ( \Delta ) | Determines the nature of the roots | If ( \Delta > 0 ), two distinct real roots; if ( \Delta = 0 ), one real root; if ( \Delta < 0 ), no real roots |
| Roots | The values of ( x ) where ( ax^2 + bx + c = 0 ) | The sign of the quadratic changes at the roots if they are real |
Formulas Involving the Sign of Quadratic Equations
The sign of a quadratic expression can be analyzed using the following formulas:
- Vertex Form: The quadratic equation can be written in vertex form as:
$$ y = a(x - h)^2 + k $$
where ( (h, k) ) is the vertex of the parabola. The sign of ( y ) depends on the sign of ( a ) and the distance of ( x ) from ( h ).
- Factored Form: If the quadratic equation can be factored, it can be written as:
$$ y = a(x - r_1)(x - r_2) $$
where ( r_1 ) and ( r_2 ) are the roots of the equation. The sign of ( y ) can be determined based on the values of ( x ) relative to ( r_1 ) and ( r_2 ).
Examples to Explain the Important Points
Example 1: Positive Leading Coefficient
Consider the quadratic equation:
$$ y = x^2 - 4x + 3 $$
The leading coefficient ( a = 1 ) is positive, so the parabola opens upwards. The discriminant is ( \Delta = (-4)^2 - 4(1)(3) = 4 ), which is positive, indicating two distinct real roots. Factoring the equation gives:
$$ y = (x - 1)(x - 3) $$
The roots are ( x = 1 ) and ( x = 3 ). The sign of ( y ) will be positive when ( x < 1 ) or ( x > 3 ), and negative when ( 1 < x < 3 ).
Example 2: Negative Leading Coefficient
Consider the quadratic equation:
$$ y = -x^2 + 6x - 9 $$
The leading coefficient ( a = -1 ) is negative, so the parabola opens downwards. The discriminant is ( \Delta = (6)^2 - 4(-1)(-9) = 0 ), indicating one real root. Completing the square gives:
$$ y = -(x - 3)^2 $$
The root is ( x = 3 ). The sign of ( y ) will be negative for all ( x ) except at ( x = 3 ), where ( y = 0 ).
Example 3: Complex Roots
Consider the quadratic equation:
$$ y = x^2 + x + 1 $$
The leading coefficient ( a = 1 ) is positive, so the parabola opens upwards. The discriminant is ( \Delta = (1)^2 - 4(1)(1) = -3 ), which is negative, indicating no real roots. The quadratic expression is always positive for all real values of ( x ).
By understanding the sign of the leading coefficient, the discriminant, and the roots of a quadratic equation, one can determine the sign of the quadratic expression for any value of ( x ). This knowledge is crucial for solving inequalities, optimizing functions, and analyzing the behavior of quadratic models in various applications.