Quadratic Formula Calculator

x = (−b ± √(b² − 4ac)) / 2a

Discriminant (b² − 4ac):
  > 0  →  two distinct real roots
  = 0  →  one repeated real root
  < 0  →  two complex roots

The ± means you calculate two values: one with addition and one with subtraction. These give you both x-intercepts of the parabola y = ax² + bx + c.

Root 1 (a in A1, b in B1, c in C1)

=(-B1+SQRT(B1^2-4*A1*C1))/(2*A1)

Root 2

=(-B1-SQRT(B1^2-4*A1*C1))/(2*A1)

Discriminant only

=B1^2-4*A1*C1

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Where the Quadratic Formula Comes From

The quadratic formula is derived by completing the square on the general quadratic equation ax² + bx + c = 0. Starting with ax² + bx + c = 0, divide by a, move c to the right side, add (b/2a)² to both sides to complete the square, factor the left side as a perfect square, take the square root of both sides, and isolate x. The result is the familiar x = (-b ± √(b²-4ac)) / 2a. Understanding this derivation means you could reconstruct the formula from scratch if you ever forgot it — a much stronger form of knowledge than rote memorization.

The ± symbol in the formula is why quadratics have potentially two solutions. When you take the square root of both sides, both the positive and negative root are valid. These correspond to the two x-intercepts of the parabola y = ax² + bx + c.

The Discriminant: Predicting the Nature of Roots

The discriminant (b² - 4ac) sits inside the square root of the formula and determines what type of solutions exist before you calculate them. When positive, there are two distinct real roots — the parabola crosses the x-axis at two points. When zero, there is exactly one real root (a repeated root) — the parabola just touches the x-axis at its vertex. When negative, the square root produces an imaginary number, yielding two complex conjugate roots — the parabola doesn't touch the x-axis at all.

Checking the discriminant first is a useful habit: it lets you know what to expect and catch errors. If the discriminant is negative but you expected real solutions, something went wrong in setting up the equation.

When to Use the Quadratic Formula vs Other Methods

Factoring is faster when integer roots are obvious: x² - 5x + 6 = (x-2)(x-3) can be spotted quickly. The quadratic formula works universally — for any quadratic regardless of whether clean integer roots exist. Completing the square is useful when you need vertex form for graphing, or when deriving related results. For a guaranteed, mechanical solution to any quadratic, the formula is the right choice.

One caveat: for very large or very small coefficients, numerical errors can affect calculator results. For hand calculation, always simplify the discriminant before taking the square root, and check answers by substitution: plug your solutions back into the original equation to verify they satisfy it.

Real-World Applications

Quadratic equations model any situation where a rate changes proportionally with time or quantity: projectile motion (height = -16t² + v₀t + h₀), profit optimization (where revenue and cost are both linear functions of quantity, their difference is quadratic), area problems (a rectangle with perimeter 40 has area = l × (20-l), a quadratic in l), and electrical circuit analysis. The quadratic formula is the bridge between the model and the solution.

Frequently Asked Questions

The discriminant is b²-4ac. Positive = two real roots. Zero = one repeated root. Negative = two complex roots (no real solution, parabola does not cross x-axis).

If a=0, the equation becomes linear (bx + c = 0), not quadratic. The formula divides by 2a, which would be division by zero.

Factoring is faster when integer roots are obvious. The quadratic formula always works for any quadratic, so it's the safe choice when factoring isn't immediately obvious.

When roots are complex, the parabola y = ax² + bx + c never crosses the x-axis — it sits entirely above or below it. Complex roots always appear in conjugate pairs (a+bi and a-bi).