Quadratic Equation Solver
Use this calculator to find the roots (solutions) of any quadratic equation in the standard form: ax² + bx + c = 0.
Understanding Quadratic Equations
A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which the unknown variable is raised to the power of two. The standard form of a quadratic equation is:
ax² + bx + c = 0
Where:
xrepresents the unknown variable.a,b, andcare coefficients, withanot equal to zero.
The "roots" or "solutions" of a quadratic equation are the values of x that satisfy the equation, making it true. Graphically, these are the x-intercepts where the parabola (the graph of a quadratic function) crosses the x-axis.
The Quadratic Formula
The most common method to solve quadratic equations is using the quadratic formula:
x = [-b ± √(b² - 4ac)] / 2a
The term inside the square root, (b² - 4ac), is called the discriminant (often denoted by Δ or D). The value of the discriminant tells us about the nature of the roots:
- If
D > 0: There are two distinct real roots. The parabola intersects the x-axis at two different points. - If
D = 0: There is exactly one real root (a repeated root). The parabola touches the x-axis at exactly one point (its vertex). - If
D < 0: There are two complex conjugate roots. The parabola does not intersect the x-axis.
How to Use the Calculator
To use the calculator, simply input the coefficients a, b, and c from your quadratic equation into the respective fields. For example, if your equation is 2x² + 5x - 3 = 0, you would enter 2 for 'a', 5 for 'b', and -3 for 'c'. Click "Calculate Roots" to see the solutions.
Examples
Example 1: Two Distinct Real Roots
Equation: x² - 3x + 2 = 0
- a = 1
- b = -3
- c = 2
Discriminant (D) = (-3)² – 4(1)(2) = 9 – 8 = 1
Since D > 0, there are two distinct real roots:
x = [3 ± √1] / 2(1)
x1 = (3 + 1) / 2 = 4 / 2 = 2
x2 = (3 – 1) / 2 = 2 / 2 = 1
(Try these values in the calculator: a=1, b=-3, c=2)
Example 2: One Real Root (Repeated)
Equation: x² - 4x + 4 = 0
- a = 1
- b = -4
- c = 4
Discriminant (D) = (-4)² – 4(1)(4) = 16 – 16 = 0
Since D = 0, there is one real root:
x = [4 ± √0] / 2(1)
x = 4 / 2 = 2
(Try these values in the calculator: a=1, b=-4, c=4)
Example 3: Two Complex Conjugate Roots
Equation: x² + 2x + 5 = 0
- a = 1
- b = 2
- c = 5
Discriminant (D) = (2)² – 4(1)(5) = 4 – 20 = -16
Since D < 0, there are two complex conjugate roots:
x = [-2 ± √-16] / 2(1)
x = [-2 ± 4i] / 2
x1 = -1 + 2i
x2 = -1 – 2i
(Try these values in the calculator: a=1, b=2, c=5)