Quadratic Equation Solver
Visually explore the equation 1x² - 3x + 2 = 0
Quadratic Equation Study Guide & FAQs
What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation in a single variable `x`. The standard form is:
Where:
xis the variable.a,b, andcare known coefficients (numbers).- The coefficient
acannot be equal to 0. If `a=0`, the equation becomes a linear equation, not a quadratic one.
The graph of a quadratic equation is a U-shaped curve called a parabola. Solving the equation means finding the points where this parabola intersects the x-axis. These points are called the roots or solutions of the equation.
The Quadratic Formula
The most reliable way to find the roots of any quadratic equation is the quadratic formula:
Understanding the Discriminant (Δ)
The part of the formula inside the square root, b2 - 4ac, is called the discriminant (Δ). It tells you about the nature of the roots without fully solving the equation:
- If Δ > 0 (positive): There are two distinct real roots. The parabola intersects the x-axis at two different points.
- If Δ = 0: There is exactly one real root (a repeated root). The parabola's vertex touches the x-axis.
- If Δ < 0 (negative): There are two complex roots. The parabola does not intersect the x-axis.
Frequently Asked Questions
If 'a' were 0, the `ax²` term would disappear, and the equation would become `bx + c = 0`. This is a linear equation, not a quadratic one. The defining characteristic of a quadratic equation is its second-degree (`x²`) term.
When the discriminant is negative, you have to find the square root of a negative number, which isn't possible in the real number system. Complex numbers use the imaginary unit `i` (where i = √-1) to express these roots. In terms of the graph, complex roots mean the parabola never touches or crosses the x-axis.
The vertex is the highest or lowest point of the parabola. If 'a' is positive, the parabola opens upwards and the vertex is the minimum point. If 'a' is negative, it opens downwards and the vertex is the maximum point. The x-coordinate of the vertex is always located at `x = -b / 2a`.
They are used in many fields, including physics (to model the path of a projectile), engineering (to design curved structures like bridges), and business (to find maximum profit or minimum cost).
