What next?
Roots of a quadratic, then the wider algebra around them.
What your result means
The roots are the values of x where the parabola crosses the x-axis. The discriminant decides how many there are before you calculate anything: positive gives two, zero gives one repeated root where the curve just touches the axis, and negative gives a pair of complex conjugates because the curve never reaches the axis at all.
Why this one is different
The discriminant appears as a number and is then translated: two real roots, one repeated root, or a complex pair. Vertex and axis of symmetry come with it for sketching, and the sum and product of the roots are printed as a check on the arithmetic you can verify in a line.
The discriminant tells you the answer before you solve
One subtraction — b² − 4ac — reveals the entire character of the solution. In exam conditions and in engineering work alike, that shortcut saves you from grinding through a formula only to discover the answer is complex.
The vertex is the other free insight. It sits at x = −b ÷ 2a, exactly halfway between the two roots when they exist, and gives the maximum or minimum value of the whole expression.
How it works
The quadratic formula comes from completing the square on ax² + bx + c = 0. The ± in the formula produces the two roots, and the square root of the discriminant determines whether those roots are real or complex. The vertex is found by setting the derivative to zero, which lands at x = −b ÷ 2a.
How to use this calculator
- Rearrange your equation into the form ax² + bx + c = 0 — everything on one side.
- Enter a, b and c, including any minus signs.
- Read the roots, then check the discriminant to understand why there are one, two or none.
- Use the vertex when you need the maximum or minimum rather than the roots.
Formula
The discriminant Δ = b² − 4ac. Vertex at x = −b ÷ 2a. Sum of roots = −b ÷ a; product of roots = c ÷ a.
Example calculation
Solving x² − 5x + 6 = 0:
x = (5 ± 1) ÷ 2
x = 3 or x = 2
Vertex at (2.5, −0.25)
Frequently asked questions
What if a is zero?＋
Then there is no x² term and the equation is linear, not quadratic. Use the algebra calculator to solve bx + c = 0 instead.
What do complex roots mean in practice?＋
They mean the parabola never touches the x-axis. In physical problems that usually signals no real-world solution exists — a projectile that never reaches a given height, for instance — though in electrical engineering complex roots describe oscillating behaviour directly.
Why does factorising sometimes fail?＋
Factorising only works cleanly when the roots are rational. The quadratic formula works for every case, which is why it is the reliable fallback when a factorisation is not obvious.
What does the discriminant tell me?＋
The value of b squared minus 4ac. Positive means two distinct real roots, zero means one repeated root, and negative means the roots are complex and the parabola never crosses the x axis.
How do I find the vertex?＋
The turning point sits at x equals minus b over 2a. Substituting that back into the equation gives the y value, which is the minimum when a is positive and the maximum when it is negative.
When should I complete the square instead?＋
When you need the vertex form, or when you want to see the transformation of the parabola. The formula is faster for roots; completing the square is more informative about shape.
Where do quadratics turn up outside a textbook?＋
Projectile paths under gravity, the area of a rectangle with a fixed perimeter, braking distance against speed, and profit against price. Any relationship with a single turning point is a candidate.
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Assumptions & limitations
Every figure here comes from a simplified model. Keep these limits in mind when reading your result:
- Handles second-degree equations in a single variable only.
- Results are shown to four decimal places; irrational roots are approximations.
- Complex roots are given in a ± bi form.