Quadratic Formula Calculator
x = 4, x = -1
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The Quadratic Formula
For any quadratic equation in standard form, ax² + bx + c = 0 (with a ≠ 0), the solutions are given by the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). The expression under the square root, b² - 4ac, is called the discriminant, and its sign alone tells you what kind of roots the equation has before you even finish solving it — positive means two distinct real roots, zero means exactly one repeated real root, and negative means two complex conjugate roots.
What the Discriminant Tells You
A common misconception is that a negative discriminant means "no solution." It actually means there's no real solution — the two roots still exist as a complex conjugate pair, x = (-b/2a) ± (√|discriminant|/2a)i. Graphically, the discriminant's sign describes how the parabola y = ax² + bx + c relates to the x-axis: two crossings, one tangent touch, or no crossing at all.
Vertex Form and the Axis of Symmetry
Every parabola is symmetric about the vertical line x = -b/(2a), called the axis of symmetry, and the vertex itself sits at (-b/(2a), c - b²/(4a)) — which is exactly the midpoint between the two roots when real roots exist. If you need to factor the same expression instead of using the formula, the factor calculator can help, and for a broader range of algebraic and scientific operations see the scientific calculator.
Frequently Asked Questions
What does a negative discriminant mean?
It means the equation has no real solutions, but it still has two solutions in the complex numbers, expressed as a conjugate pair x = (-b/2a) plus-or-minus (sqrt(|discriminant|)/2a)i. Graphically, the parabola never crosses the x-axis.
What happens if I enter 0 for a?
With a = 0 the equation is no longer quadratic. The calculator falls back to solving the resulting linear equation bx + c = 0 (giving x = -c/b), or reports no solution / infinite solutions if b is also 0.