Quadratic Equation Solver
Print| Discriminant (D) | 16 |
| Roots Type | Two Real Roots |
| Parabola Vertex (h, k) | (-1, -4) |
| Y-Intercept | -3 |
In algebra, a quadratic equation is a polynomial equation of the second degree. The standard mathematical form is:
ax^2 + bx + c = 0
Where x is the unknown variable, a is the quadratic coefficient (which cannot equal 0, or the equation becomes linear), b is the linear coefficient, and c is the constant term. The coefficients represent known real numbers.
Our free Quadratic Formula Calculator solves standard second-degree equations instantly. It displays the real or complex (imaginary) roots of the equation, identifies the discriminant, and plots the parabolic axis of symmetry. The tool supports integers, decimals, and fractional inputs (like 1/4 or -3/2).
The Quadratic Formula & Roots
While quadratic equations can be solved via factoring, graphing, or completing the square, the most robust method is using the quadratic formula:
x = [ -b ± √(b^2 - 4ac) ] / 2a
The symbol ± (plus-minus) indicates that quadratic equations typically have two solutions. Geometrically, these roots represent the x-coordinates where the corresponding parabola y = ax^2 + bx + c crosses the x-axis (its x-intercepts).
The Discriminant (b^2 – 4ac)
The term inside the square root, b^2 - 4ac, is called the discriminant (D). It determines the number and nature of the roots:
- D > 0: The equation has two distinct real roots (the parabola crosses the x-axis in two places).
- D = 0: The equation has one real double root (the parabola’s vertex touches the x-axis at exactly one point).
- D < 0: The equation has two complex (imaginary) roots (the parabola never touches the x-axis; the roots involve the imaginary unit i where
i = √-1).
Step-by-Step Derivation (Completing the Square)
The quadratic formula is derived directly by completing the square on the general equation ax^2 + bx + c = 0:
- Start with:
ax^2 + bx + c = 0 - Divide the entire equation by the quadratic coefficient
a:x^2 + (b/a)x + (c/a) = 0 - Move the constant term to the right side:
x^2 + (b/a)x = -(c/a) - Complete the square on the left side by adding the square of half the linear coefficient,
(b / 2a)^2, to both sides:x^2 + (b/a)x + (b / 2a)^2 = -(c/a) + (b / 2a)^2 - Factor the perfect square trinomial on the left and simplify the fraction on the right:
(x + b/2a)^2 = -(c/a) + b^2/4a^2(x + b/2a)^2 = -4ac/4a^2 + b^2/4a^2 = (b^2 - 4ac) / 4a^2 - Take the square root of both sides (incorporating the plus-minus sign):
x + b/2a = ± √(b^2 - 4ac) / 2a - Subtract
b/2afrom both sides to solve for x:x = [ -b ± √(b^2 - 4ac) ] / 2a
Parabolic Symmetry & Physics Applications
The quadratic formula also provides the equation for the parabola’s axis of symmetry. The vertical line that splits the parabola symmetrically down the center is given by:
x = -b / 2a
This value represents the x-coordinate of the parabola’s vertex (its absolute minimum or maximum peak).
Quadratic equations model many real-world phenomena:
- Projectile Trajectories: Calculating the height, distance, and landing time of objects thrown or launched under the influence of gravity.
- Area Optimization: Finding the maximum dimensions of a rectangular boundary under fixed perimeter constraints.
- Motion and Speed: Solving kinematics models where distance is proportional to the square of time (
d = vt + 0.5at^2).
Review standard geometric shapes using our Triangle Calculator or solve algebraic power ratios with the Exponent Calculator.
Frequently Asked Questions (FAQ)
What is a quadratic equation?
A quadratic equation is a second-degree polynomial equation where the highest exponent of the variable (x) is 2. The general form is ax^2 + bx + c = 0, where a, b, and c are numbers, and a is not equal to 0.
Can a quadratic equation have zero roots?
A quadratic equation always has exactly two roots. However, depending on the discriminant, these roots may be real numbers or complex (imaginary) numbers. If the parabola does not cross the x-axis, the roots are complex.
How do you find the vertex of a quadratic function?
The x-coordinate of the vertex lies on the axis of symmetry, calculated as h = -b / 2a. To find the y-coordinate of the vertex, plug this x-value back into the original quadratic function: k = a(h)^2 + b(h) + c.
Why can the coefficient “a” never be zero?
If a = 0, the ax^2 term disappears, reducing the equation to bx + c = 0. This is a first-degree linear equation, which produces a straight line on a graph rather than a curved parabola.