Characteristics of quadratic equations

Key Notes :

Quadratic Equations

A quadratic equation is a polynomial equation of degree two. It can be expressed in the general form:

ax² + bx + c = 0

where a, b, and c are constants, and a ≠ 0.

Key Characteristics:

1. Degree: The highest power of the variable (x) in a quadratic equation is 2.
2. Roots: A quadratic equation has at most two solutions, also known as roots. These roots represent the values of x that make the equation true.
3. Nature of Roots:
   • Real and Distinct: If the discriminant (b² – 4ac) is positive, the roots are real and distinct.
   • Real and Equal: If the discriminant is zero, the roots are real and equal.
   • Imaginary or Complex: If the discriminant is negative, the roots are imaginary or complex.
4. Sum and Product of Roots:
   • Sum of Roots: The sum of the roots of a quadratic equation is -b/a.
   • Product of Roots: The product of the roots of a quadratic equation is c/a.
5. Graph: The graph of a quadratic equation is a parabola.
   • Shape: The shape of the parabola depends on the sign of the coefficient a.
     • If a > 0, the parabola opens upward.
     • If a < 0, the parabola opens downward.
   • Vertex: The vertex of the parabola is the point where the equation changes direction.
6. Completing the Square: This is a technique used to solve quadratic equations by rewriting them in the form (x + h)² = k.
7. Quadratic Formula: The quadratic formula is a general formula used to solve quadratic equations:
   • x = [-b ± √(b² – 4ac)] / 2a

Additional Points to Consider:

• Factoring: In some cases, quadratic equations can be solved by factoring.
• Discriminant: The discriminant (b² – 4ac) provides information about the nature of the roots.
• Graphing: Understanding the graph of a quadratic equation can help visualize the solutions.
• Applications: Quadratic equations have many applications in various fields, including physics, engineering, and economics.

Upward-Facing Parabola

A parabola opens upward when the leading coefficient (a) is positive. This means that the quadratic term, ax², has a positive sign.

Key characteristics of an upward-facing parabola:

• Vertex: The lowest point on the parabola, also known as the minimum point.
• Axis of symmetry: A vertical line that divides the parabola into two symmetrical halves.
• X-intercepts: The points where the parabola intersects the x-axis.
• Y-intercept: The point where the parabola intersects the y-axis.

Example equation:

• y = x²

In this equation, the leading coefficient is 1, which is positive, resulting in an upward-facing parabola.

Downward-Facing Parabola

A parabola opens downward when the leading coefficient (a) is negative. This means that the quadratic term, ax², has a negative sign.

Key characteristics of a downward-facing parabola:

• Vertex: The highest point on the parabola, also known as the maximum point.
• Axis of symmetry: A vertical line that divides the parabola into two symmetrical halves.
• X-intercepts: The points where the parabola intersects the x-axis.
• Y-intercept: The point where the parabola intersects the y-axis.

Example equation:

• y = -x²

In this equation, the leading coefficient is -1, which is negative, resulting in a downward-facing parabola.

Learn with an example

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