Quadratic Equation Solver
Enter the coefficients (a, b, c) of your quadratic equation in the form ax² + bx + c = 0 to find its roots.
Understanding Quadratic Equations and Their Solutions
A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually 'x') is 2. It is typically written in the standard form:
ax² + bx + c = 0
where 'a', 'b', and 'c' are coefficients, and 'a' cannot be zero. If 'a' were zero, the equation would become a linear equation (bx + c = 0).
The Role of Coefficients
- Coefficient 'a': Determines the concavity of the parabola (the graph of a quadratic equation). If 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards.
- Coefficient 'b': Influences the position of the vertex of the parabola.
- Coefficient 'c': Represents the y-intercept of the parabola (where the graph crosses the y-axis).
Finding the Roots (Solutions)
The "roots" or "solutions" of a quadratic equation are the values of 'x' that satisfy the equation, i.e., make the equation true. Graphically, these are the points where the parabola intersects the x-axis. There can be two distinct real roots, one real root (a repeated root), or two complex conjugate roots.
The most common method to find these roots is using the quadratic formula:
x = [-b ± sqrt(b² - 4ac)] / (2a)
The Discriminant (b² – 4ac)
The term inside the square root, b² - 4ac, is called the discriminant (often denoted by the Greek letter Delta, Δ). Its value tells us about the nature of the roots:
- If Δ > 0: There are two distinct real roots. The parabola crosses the x-axis at two different points.
- If Δ = 0: There is exactly one real root (a repeated root). The parabola touches the x-axis at exactly one point (its vertex).
- If Δ < 0: There are two complex conjugate roots. The parabola does not intersect the x-axis.
Examples of Quadratic Equations:
- Example 1 (Two Real Roots):
x² - 3x + 2 = 0- Here, a=1, b=-3, c=2.
- Discriminant = (-3)² – 4(1)(2) = 9 – 8 = 1 (which is > 0).
- Roots are x = [3 ± sqrt(1)] / 2 = (3 ± 1) / 2. So, x1 = 2, x2 = 1.
- Example 2 (One Real Root):
x² - 4x + 4 = 0- Here, a=1, b=-4, c=4.
- Discriminant = (-4)² – 4(1)(4) = 16 – 16 = 0.
- Root is x = [4 ± sqrt(0)] / 2 = 4 / 2 = 2.
- Example 3 (Two Complex Roots):
x² + x + 1 = 0- Here, a=1, b=1, c=1.
- Discriminant = (1)² – 4(1)(1) = 1 – 4 = -3 (which is < 0).
- Roots are x = [-1 ± sqrt(-3)] / 2 = [-1 ± i*sqrt(3)] / 2. So, x1 = -0.5 + 0.866i, x2 = -0.5 – 0.866i.
Use the calculator above to quickly find the roots for any quadratic equation by simply entering its coefficients.