What is the quadratic formula?

The quadratic formula solves ax²+bx+c=0: x = (-b ± √(b²-4ac)) / (2a). The two solutions are x₁ = (-b + √(b²-4ac)) / (2a) and x₂ = (-b - √(b²-4ac)) / (2a). The expression b²-4ac is called the discriminant and determines whether roots are real or complex.

What is the discriminant?

The discriminant is D = b²-4ac. If D > 0: two distinct real roots. If D = 0: one repeated real root (the parabola touches the x-axis). If D < 0: two complex conjugate roots (no real roots — parabola doesn't cross x-axis).

How do you find the vertex of a parabola?

For y = ax²+bx+c, the vertex is at x = -b/(2a) and y = c - b²/(4a). The axis of symmetry is the vertical line x = -b/(2a). If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum.

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Quadratic Formula Calculator

Solve any quadratic equation ax²+bx+c=0. Get real & complex roots, discriminant, vertex, parabola graph and full step-by-step solution.

ax² + bx + c = 0

Quadratic formula

x = (−b ± √(b² − 4ac)) / (2a)

Enter coefficients

a (x² coefficient)

Must not be 0

b (x coefficient)

c (constant)

Quick examples

Discriminant D = b²−4ac —

Solutions (roots)

x₁ —

x₂ —

—Vertex (h, k)

—Axis of symmetry

—Opens

—y-intercept

📈 Parabola graph

📝 Step-by-step solution

How the quadratic formula works

Any equation of the form ax²+bx+c=0 (where a≠0) is a quadratic. The quadratic formula always works and gives exact solutions, unlike factoring (which only works for "nice" integers) or completing the square (which can be tedious).

🟢 D > 0: Two real roots

When b²−4ac > 0, the parabola crosses the x-axis at two distinct points. Both roots are real numbers. If D is a perfect square, the roots are rational (the equation can be factored).

🟡 D = 0: One repeated root

When b²−4ac = 0, the parabola touches the x-axis at exactly one point — the vertex. The single root x = −b/(2a) has multiplicity 2. The equation is a perfect square: a(x−r)²=0.

🟣 D < 0: Complex roots

When b²−4ac < 0, the parabola never crosses the x-axis. Roots are complex conjugates: x = (−b ± i√|D|) / (2a). They come in conjugate pairs: if p+qi is a root, so is p−qi.

📌 Vertex form

y = a(x−h)²+k where vertex (h,k): h = −b/(2a), k = c − b²/(4a). Useful for graphing. If a>0, vertex is minimum; if a<0, vertex is maximum. The axis of symmetry is x = h.

Frequently asked questions

How do I use the quadratic formula step by step?

1) Identify a, b, c from ax²+bx+c=0. 2) Calculate the discriminant: D=b²−4ac. 3) If D≥0: x = (−b±√D) / (2a). 4) Calculate x₁=(−b+√D)/(2a) and x₂=(−b−√D)/(2a). Example: x²−5x+6=0 → a=1,b=−5,c=6 → D=25−24=1 → x=(5±1)/2 → x₁=3, x₂=2.

What are Vieta's formulas?

For roots x₁ and x₂ of ax²+bx+c=0: Sum of roots: x₁+x₂ = −b/a. Product of roots: x₁×x₂ = c/a. These let you check your answers: if roots are 2 and 3, sum=5=−(−5)/1 ✓, product=6=6/1 ✓. You can also reconstruct the equation from roots: a(x−x₁)(x−x₂)=0.

When should I use factoring vs the quadratic formula?

Factoring is faster when it works — look for integer roots by testing factors of c/a. The quadratic formula always works for any coefficients, including decimals and fractions. Completing the square is useful for deriving vertex form. The quadratic formula is the most reliable all-purpose method.

What if a = 0?

If a=0, the equation is linear (bx+c=0), not quadratic. The "quadratic formula" requires a≠0 because we divide by 2a. Linear equations have one solution: x=−c/b (if b≠0). If both a and b are 0, the equation has either no solution (c≠0) or infinitely many solutions (c=0).