Quadratic Equation Calculator

A quadratic equation is a second-degree polynomial equation in the form:

$ax^2 + bx + c = 0$

where $a$, $b$, and $c$ are constants and $a \neq 0$.

The graph of a quadratic equation is a parabola — a U-shaped curve that opens upward when $a > 0$ and downward when $a < 0$. The solutions (also called roots or zeros) are the x-values where the parabola crosses the x-axis.

There are four main methods:

1. Quadratic Formula

The most universal method. For $ax^2 + bx + c = 0$:

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

The discriminant $\Delta = b^2 - 4ac$ determines the number of solutions:

• $\Delta > 0$: two distinct real roots
• $\Delta = 0$: one repeated real root
• $\Delta < 0$: two complex conjugate roots

2. Factoring

If the quadratic can be expressed as $(x - r_1)(x - r_2) = 0$, the roots are $r_1$ and $r_2$.

Example: $x^2 + 5x + 6 = (x+2)(x+3) = 0$ → $x = -2$ or $x = -3$

3. Completing the Square

Rewrite $ax^2 + bx + c = 0$ into $(x + p)^2 = q$ form, then solve by taking square roots.

4. Graphing

Plot $y = ax^2 + bx + c$ and find the x-intercepts.

• Forgetting that $a \neq 0$: If $a = 0$, it becomes a linear equation.
• Sign errors in the formula: Be careful with $-b$ — if $b$ is negative, $-b$ is positive.
• Forgetting the $\pm$: The formula gives two solutions. Don't drop one.
• Not simplifying radicals: Always simplify $\sqrt{b^2 - 4ac}$ as much as possible.
• Division errors: Remember to divide the entire numerator by $2a$.