Quadratic Equation Solver

A quadratic equation is a second-order polynomial equation in a single variable $x$, with a non-zero coefficient for $x^2$. The general form is:

$ax^2 + bx + c = 0$

Where $x$ represents an unknown, and $a$, $b$, and $c$ represent known numbers (coefficients), such that $a$ is not equal to zero. If $a = 0$, the equation becomes linear ($bx + c = 0$), and it is no longer quadratic.

What is the Discriminant?

The term $b^2 - 4ac$ is known as the discriminant (often denoted by the Greek letter delta, $\Delta$). It is the part of the quadratic formula under the square root sign. The value of the discriminant determines the nature of the roots:

1. Positive Discriminant ($\Delta > 0$): The equation has two distinct real roots. The parabola crosses the x-axis at two points.
2. Zero Discriminant ($\Delta = 0$): The equation has exactly one real root (a repeated root). The vertex of the parabola touches the x-axis.
3. Negative Discriminant ($\Delta < 0$): The equation has two complex (imaginary) roots. The parabola never touches or crosses the x-axis.

The Quadratic Formula

To solve for $x$, we use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This formula works for all quadratic equations, whether the roots are real, rational, irrational, or complex.

How to Use This Calculator

1. Enter Coefficient 'a': This is the number before the $x^2$ term. It cannot be zero.
2. Enter Coefficient 'b': This is the number before the $x$ term.
3. Enter Coefficient 'c': This is the constant term at the end.
4. Review Results: The calculator will immediately display the discriminant, the type of roots, the roots themselves, and the coordinates of the vertex (the peak or valley of the parabola).
5. Study the Steps: Scroll down to see the step-by-step algebraic derivation of your specific solution.

Worked Examples

Example 1: Two Real Roots

Solve: $x^2 - 5x + 6 = 0$

• $a = 1, b = -5, c = 6$
• $\Delta = (-5)^2 - 4(1)(6) = 25 - 24 = 1$
• Since $\Delta > 0$, we have two roots:
• $x_1 = (5 + \sqrt{1}) / 2 = 3$
• $x_2 = (5 - \sqrt{1}) / 2 = 2$

Example 2: Complex Roots

Solve: $x^2 + x + 1 = 0$

• $a = 1, b = 1, c = 1$
• $\Delta = (1)^2 - 4(1)(1) = -3$
• Since $\Delta < 0$, roots are complex:
• $x = \frac{-1 \pm i\sqrt{3}}{2} = -0.5 \pm 0.866i$

FAQ

Can 'a' be zero?

No. If $a = 0$, the $x^2$ term disappears, leaving $bx + c = 0$, which is a linear equation. Quadratic equations must have a squared term.

What is the vertex of a parabola?

The vertex is the highest or lowest point on the graph of a quadratic function. For the equation $y = ax^2 + bx + c$, the x-coordinate of the vertex is found using $x = -b / 2a$.

What if 'b' or 'c' are zero?

That's perfectly fine. If $b=0$, the equation is $ax^2 + c = 0$ (pure quadratic). If $c=0$, the equation is $ax^2 + bx = 0$, which can be solved easily by factoring out an $x$.

Why are complex roots always in pairs?

Due to the Conjugate Root Theorem, if a polynomial with real coefficients has a complex root $a + bi$, then its conjugate $a - bi$ must also be a root.

Does the calculator show the graph?

Yes, the calculator generates a plot of the parabola centered around its vertex so you can visually see where it crosses the x-axis (the roots) and the y-axis (the constant $c$).

How does the discriminant relate to the graph?

If the discriminant is positive, the graph has two x-intercepts. If zero, one x-intercept. If negative, zero x-intercepts.

Limitations

This calculator assumes real-numbered coefficients. It does not support solving equations where the coefficients $a, b,$ or $c$ are themselves complex numbers or matrices.