Quadratic Formula Calculator
Enter the coefficients of ax² + bx + c = 0 and get both roots — real or complex — plus the discriminant that decides which case you are in, the vertex of the parabola, and the factored form when the equation factorizes.
Matched against the form a(x − r₁)(x − r₂) with r₁ + r₂ = −b ÷ a and r₁ × r₂ = c ÷ a.
Formula and calculation
Worked example
For x² − 3x + 2 = 0 the coefficients are a = 1, b = −3 and c = 2, so the discriminant is 1. Because it is positive and a perfect square the equation factorizes: (x - 2)(x - 1), giving the roots 2 and 1. The vertex of the parabola sits at (1.5, -0.25).
What the discriminant tells you before you calculate
Δ = b² − 4ac is the part under the square root, and it is the quickest way to know what kind of answer to expect. A positive Δ means the parabola crosses the x-axis twice, so there are two different real roots. Δ = 0 means the vertex sits exactly on the axis and both roots collapse into one number — the equation is a perfect square. A negative Δ means the parabola never touches the axis, so the roots are complex: they have a real part (−b ÷ 2a) and an imaginary part (√(−Δ) ÷ 2a), and they always come in a conjugate pair, a + bi and a − bi. Checking Δ first also catches mistakes: if your formula gives a positive number where Δ is negative, something slipped.
Factoring versus the formula, and Vieta's shortcut
When the roots are whole numbers or simple fractions, factoring is faster: x² − 3x + 2 is (x − 1)(x − 2) after a moment's thought, and the roots fall out immediately. The formula never needs that moment — it works for every quadratic, including ones with √5 in the answer. A useful check on either method is Vieta's relations: the two roots must add up to −b ÷ a and multiply to c ÷ a. For x² − 3x + 2 that means 1 + 2 = 3 and 1 × 2 = 2, which both hold. If your roots fail those two tests, the arithmetic went wrong somewhere.
What the results mean
- First root
- One of the two solutions of the equation. When the discriminant is negative the roots are complex numbers written in the form a + bi, where i is the square root of −1.
- Second root
- The other solution. Real roots come out of the same formula with a minus where root1 has a plus; complex roots are the conjugate pair a + bi and a − bi.
- Discriminant
- The value Δ = b² − 4ac, which decides the case: positive gives two different real roots, zero gives one repeated root, negative gives a pair of complex roots.
- Vertex
- The turning point (h, k) of the parabola y = ax² + bx + c, written as a coordinate pair. h = −b ÷ 2a is where the axis of symmetry sits, and k is the highest or lowest value the parabola reaches.
- Parabola opens
- Whether the parabola opens upward (a positive, so the vertex is a minimum) or downward (a negative, so the vertex is a maximum).
- Factored form
- The equation rewritten as a product of factors, for example (x − 1)(x − 2), when the roots are rational numbers. If the roots are irrational or complex, no whole-number factorization exists and the row says so.
- Calculation
- The quadratic formula with your coefficients substituted in, so you can follow the arithmetic line by line.
Frequently asked questions
How do I know if a quadratic has no real solution?
Work out the discriminant Δ = b² − 4ac. If it is negative, the equation has no real roots — the parabola never crosses the x-axis. The two solutions are complex numbers instead, written as a + bi, and this calculator reports them in that form rather than saying there is no answer.
What does the discriminant tell me?
It counts the real roots. Δ > 0 gives two different real roots, Δ = 0 gives one repeated root (the equation is a perfect square), and Δ < 0 gives a pair of complex roots. For 2x² − 7x + 3, Δ = 49 − 24 = 25, so there are two real roots, 3 and 0.5.
How do I find the vertex of a parabola?
The x-coordinate is h = −b ÷ 2a — the same −b ÷ 2a that appears in the quadratic formula — and the y-coordinate is k = c − b² ÷ 4a. For x² − 3x + 2 the vertex is at (1.5, −0.25), halfway between the two roots, which is always true for a parabola.
Why does the quadratic formula have a ± sign?
Because a square root has two values, one positive and one negative, and both give a solution: the two roots differ only in whether √Δ is added or subtracted. That is also why the discriminant decides how many real roots exist — a negative number has no real square root, so the ± pair becomes complex.
Can I use this calculator if a is not 1?
Yes, and that is where the formula earns its keep. With 2x² − 7x + 3 you would divide every term by 2 first if factoring, but the formula takes the coefficients as they are: (−(−7) ± √25) ÷ 4 gives 3 and 0.5. The factored form shown is the integer one, (2x − 1)(x − 3).