Quadratic Formula Calculator – Step-by-Step Solver for ax² + bx + c = 0

Q: What is the quadratic formula used for?

The quadratic formula x = [-b ± √(b² - 4ac)] / (2a) gives all solutions of any quadratic equation ax² + bx + c = 0 with a ≠ 0. It works whether the equation factors nicely or not, and it also tells you how many real solutions exist by looking at the discriminant b² - 4ac.

Q: How do I know how many solutions a quadratic has?

The discriminant Δ = b² - 4ac controls the number and type of solutions. If Δ > 0, there are two distinct real solutions. If Δ = 0, there is exactly one real solution (a repeated root). If Δ < 0, there are no real solutions, but there are two complex conjugate solutions.

Q: What if a is zero in ax² + bx + c = 0?

If a = 0, the equation is no longer quadratic; it becomes a linear equation bx + c = 0, provided b ≠ 0. In that case, the solution is x = -c/b. If both a = 0 and b = 0, then either there is no solution (if c ≠ 0) or infinitely many solutions (if c = 0). The quadratic formula only applies when a ≠ 0.

Q: Can I rely on this calculator for exams or graded work?

This tool is designed for learning, checking homework, and building intuition. Always follow your teacher's rules for calculators and show full working when required. You are responsible for ensuring that any answers you submit meet the expectations of your course or exam.

The quadratic formula at a glance

A quadratic equation in one variable is an equation that can be written in the form

\[ ax^2 + bx + c = 0, \quad a \ne 0 \]

where \(a\), \(b\), and \(c\) are constants and \(x\) is the unknown. The quadratic formula gives the solutions:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]

The expression under the square root, \(\Delta = b^2 - 4ac\), is called the discriminant and determines the number and type of solutions.

How the discriminant tells you the type of roots

\[ \Delta = b^2 - 4ac \]

If \(\Delta > 0\): two distinct real roots.
If \(\Delta = 0\): one real repeated root (a double root).
If \(\Delta < 0\): two complex conjugate roots, no real intersection with the x-axis.

This is why the discriminant is so useful: before computing square roots, you already know what kind of solutions to expect.

Vertex and graph of a quadratic

The graph of \(y = ax^2 + bx + c\) is a parabola. The point where it turns is called the vertex, and its x-coordinate is always

\[ x_{\text{vertex}} = -\frac{b}{2a}, \qquad y_{\text{vertex}} = f(x_{\text{vertex}}) = a x_{\text{vertex}}^2 + b x_{\text{vertex}} + c. \]

The line \(x = -\dfrac{b}{2a}\) is the axis of symmetry. If \(a > 0\), the parabola opens upward (like a cup); if \(a < 0\), it opens downward.

Examples you can try

Two real roots: \(x^2 - 5x + 6 = 0\). Here \(a = 1\), \(b = -5\), \(c = 6\). The discriminant is \(\Delta = (-5)^2 - 4 \cdot 1 \cdot 6 = 25 - 24 = 1\). Roots: \(x = \dfrac{5 \pm 1}{2}\) → \(x = 2\) and \(x = 3\).
One repeated root: \(x^2 - 4x + 4 = 0\). \(\Delta = (-4)^2 - 4 \cdot 1 \cdot 4 = 16 - 16 = 0\). Only one real solution \(x = \dfrac{4}{2} = 2\), but it counts twice.
No real roots (complex): \(x^2 + 4x + 13 = 0\). \(\Delta = 4^2 - 4 \cdot 1 \cdot 13 = 16 - 52 = -36\). Roots are complex: \(x = -2 \pm 3i\).

FAQ: using the quadratic formula calculator safely

What is the quadratic formula used for?

It is a universal method to solve any quadratic equation \(ax^2 + bx + c = 0\) with \(a \ne 0\). Unlike factoring, it always works, even when the roots are irrational or complex.

How do I know how many solutions a quadratic has?

The discriminant \(\Delta = b^2 - 4ac\) tells you everything: two real solutions (\(\Delta > 0\)), one real repeated solution (\(\Delta = 0\)), or two complex solutions (\(\Delta < 0\)). This calculator shows the discriminant and explains the classification.

What if a is zero in ax² + bx + c = 0?

Then the equation is not quadratic. If \(a = 0\) but \(b \ne 0\), you have a linear equation \(bx + c = 0\) with solution \(x = -c/b\). If both \(a = 0\) and \(b = 0\), then:

if \(c \ne 0\): no solution;
if \(c = 0\): every real number is a solution.

This calculator detects the case \(a = 0\) and switches to the appropriate message.

Can I rely on this calculator for exams or graded work?

Use it as a learning companion and a quick check. For assessments, always comply with your institution’s calculator policy and show full working steps. You remain responsible for any answers you submit.

Interactive quadratic formula solver