Quadratic equation solver (ax² + bx + c = 0)

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Result

Rootsx₁ = 5/2, x₂ = −1
x₁2.5
x₂−1
Discriminant Δ49
Nature of the rootstwo different real roots
Vertex(0.75, −6.125)
Axis of symmetryx = 0.75
y-intercept−5
Parabola opensupward (minimum at the vertex)
Factored form(2x − 5)(x + 1)

Solve any quadratic equation ax² + bx + c = 0. The calculator works out the discriminant, uses the quadratic formula and gives the roots both as decimals and in exact form, as fractions or simplified square roots such as 2 ± √3, including complex roots. It also finds the vertex, the axis of symmetry and the factored form, and checks the roots with Vieta’s formulas.

Step-by-step solution

Identify a, b and c

2x² − 3x − 5 = 0: a = 2, b = −3, c = −5

Discriminant

Δ = b² − 4ac = (−3)² − 4·2·(−5) = 49
Δ > 0 → two different real roots

Quadratic formula

x = (−b ± √Δ)/(2a) = (3 ± √49)/4
exact: x₁ = 5/2, x₂ = −1
x₁ = 2.5, x₂ = −1

Vertex and axis of symmetry

h = −b/(2a) = 3/4 ≈ 0.75
k = c − b²/(4a) = −49/8 ≈ −6.125
vertex form: y = 2(x − 0.75)2 − 6.125

Factored form

2x² − 3x − 5 = (2x − 5)(x + 1)

Check with Vieta’s formulas

x₁ + x₂ = −b/a = 1.5
x₁·x₂ = c/a = −2.5

Key formulas

Quadratic with p² < 4q
z = −p/2 ± j√(q − p²/4)

How to solve a quadratic equation

  1. Write the equation in the standard form ax² + bx + c = 0 and read off a, b and c, with their signs.
  2. Calculate the discriminant Δ = b² − 4ac. If Δ > 0 there are two real roots, if Δ = 0 one repeated root, and if Δ < 0 two complex conjugate roots.
  3. Use the quadratic formula x = (−b ± √Δ)/(2a). Simplify the square root by taking out square factors: √12 = 2√3.
  4. For Δ < 0 write √Δ = i·√|Δ|, so the roots are −b/(2a) ± i·√|Δ|/(2a).
  5. The vertex of the parabola y = ax² + bx + c is at x = −b/(2a); substitute it to find the y-coordinate.
  6. Check: the roots add up to −b/a and multiply to c/a.

Questions and answers

When can I factor instead of using the formula?

When the discriminant is a perfect square and the coefficients are whole numbers, the roots are rational and the quadratic factors over the integers, as 2x² − 3x − 5 = (2x − 5)(x + 1). Otherwise the quadratic formula is the reliable method.

What does a negative discriminant mean?

The parabola does not cross the x-axis, so there are no real roots. The two roots are complex conjugates p ± qi, which matter for example in damped vibrations and in circuit analysis.

What is the vertex form?

y = a(x − h)² + k, where (h, k) is the vertex. It shows the minimum (a > 0) or maximum (a < 0) of the parabola directly and is found by completing the square.

What if a = 0?

Then the equation is linear, bx + c = 0, with the single solution x = −c/b. The quadratic formula would divide by zero, so the calculator asks for a non-zero a.

Why does the calculator give exact roots?

Exams often ask for answers such as (3 ± √17)/4 rather than decimals. When a, b and c are whole numbers or decimals, the calculator works with exact fractions and simplifies the square root.

Practise it yourself

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Practise Complex numbers

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