Quadratic Formula
Solve quadratic equations using the quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / (2a). Evaluate discriminant and roots with the math expression evaluator.
Algebra
Detailed Explanation
Quadratic Formula
The quadratic formula solves equations of the form ax^2 + bx + c = 0:
x = (-b +/- sqrt(b^2 - 4*a*c)) / (2*a)
Step-by-Step Process
For the equation 2x^2 + 5x - 3 = 0 (a=2, b=5, c=-3):
Step 1: Calculate the discriminant
b^2 - 4*a*c = 5^2 - 4*2*(-3) = 25 + 24 = 49
Step 2: Calculate the two roots
x1 = (-5 + sqrt(49)) / (2*2) = (-5 + 7) / 4 = 0.5
x2 = (-5 - sqrt(49)) / (2*2) = (-5 - 7) / 4 = -3
The Discriminant
The value b^2 - 4*a*c (called the discriminant) tells you the
nature of the roots:
| Discriminant | Meaning |
|---|---|
| > 0 | Two distinct real roots |
| = 0 | One repeated real root |
| < 0 | Two complex conjugate roots |
More Examples
x^2 - 4 = 0 (a=1, b=0, c=-4):
(0 + sqrt(0 - 4*1*(-4))) / (2*1) = sqrt(16)/2 = 2
(0 - sqrt(16)) / 2 = -2
x^2 + 2x + 1 = 0 (a=1, b=2, c=1):
(-2 + sqrt(4 - 4)) / 2 = -2/2 = -1 (repeated root)
Vertex Form
The vertex of the parabola is at:
x_vertex = -b / (2*a)
y_vertex = c - b^2 / (4*a)
Use Case
A student solving algebra homework by computing discriminants and roots step by step, or an engineer finding equilibrium points in a physical system.