The Quadratic Equation Solver Calculator solves equations of the form $ax^2 + bx + c = 0$. It uses the quadratic formula, shows whether the roots are two real, one repeated, or complex, and also provides the vertex and axis of symmetry.
This is ideal for students practising factorising and formula use, for teachers preparing examples, and for anyone who needs quick and reliable quadratic solutions.
What does the calculator do?
Solves $ax^2 + bx + c = 0$ using the quadratic formula $$
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
$$
Computes the discriminant $D = b^2 – 4ac$ to classify roots:
- $D > 0$: two distinct real roots
- $D = 0$: one repeated real root
- $D < 0$: complex conjugate roots
Handles the $a=0$ case (falls back to a linear equation $bx + c = 0$).
Shows vertex form info
Vertex $(h,k)$ where
$$
h = -\frac{b}{2a}, \quad k = f(h)
$$
- Axis of symmetry $x = h$
- Opening direction: “up” if $a>0$, “down” if $a<0$
Worked examples
Example 1 — Two real roots
Equation: $x^2 – 3x + 2 = 0$
- $a=1,;b=-3,;c=2$
- $D = (-3)^2 – 4(1)(2) = 9 – 8 = 1 > 0$
$$
x = \frac{3 \pm \sqrt{1}}{2} \;\;\Rightarrow\;\; x_1 = 2,\; x_2 = 1
$$
- Vertex:
$$
h = -\frac{-3}{2} = \frac{3}{2}, \quad k = f!\left(\frac{3}{2}\right) = -\frac{1}{4}
$$
Example 2 — Repeated root
Equation: $x^2 + 4x + 4 = 0$
- $a=1,;b=4,;c=4$
- $D = 4^2 – 16 = 0$
$$
x = \frac{-4}{2} = -2
$$
- Vertex: $(-2, 0)$
Example 3 — Complex roots
Equation: $2x^2 + x + 5 = 0$
- $a=2,;b=1,;c=5$
- $D = 1 – 40 = -39 < 0$
$$
x = \frac{-1 \pm \sqrt{-39}}{4} = \frac{-1}{4} \pm \frac{\sqrt{39}}{4}i
$$
- Vertex:
$$
h = -\frac{1}{4}, \quad k = f!\left(-\tfrac{1}{4}\right) = \frac{79}{32}
$$
Why is it important?
- In school: Quadratics connect factoring, completing the square, and graphing — foundational for algebra and precalculus.
- In real life: Used in physics (projectile motion), engineering (optimization and design), and business (profit models with parabolic behaviour).
- For STEM learners: Builds fluency with discriminants, radicals, and function forms (standard vs. vertex).
Frequently Asked Questions
Q1: What is the discriminant and why does it matter?
It’s $D = b^2 – 4ac$. It tells you the nature of the roots: positive (two real), zero (one repeated), or negative (complex).
Q2: Can this handle decimals and negatives?
Yes. Enter any real numbers for $a$, $b$, and $c$.
Q3: What if $a = 0$?
Then the quadratic becomes linear: $bx + c = 0$. The solver detects this and solves for $x$.
Q4: How do I find the vertex?
Use $h = -\dfrac{b}{2a}$ and $k = f(h)$. The calculator shows both values and the axis of symmetry.
Q5: Does the calculator factor the quadratic?
Not explicitly. If $D$ is a perfect square and coefficients are simple, the roots may be integers or simple fractions; otherwise use the formula.