AI Math Assistant
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AI provides general assistance. Always verify important calculations.
About This Calculator
The quadratic formula solves any equation of the form ax² + bx + c = 0, even when factoring by inspection fails or produces irrational roots. This calculator computes both real and complex solutions, displays the discriminant, and indicates whether roots are real and distinct, repeated, or imaginary. Quadratic equations model projectile trajectories, profit optimization curves, and countless physical phenomena involving parabolic relationships.
Quick Tips
- 1 The discriminant (b^2-4ac) tells you: positive = 2 roots, zero = 1, negative = none.
- 2 Always simplify the radical in your answer to its lowest form.
- 3 If a=0 it's not quadratic — just solve the resulting linear equation.
Example Calculation
Solve 3x^2 + 7x - 6 = 0.
x = 0.6667 (2/3) and x = -3.0 | Discriminant: 121 (two real roots) | Vertex at x = -1.17
The Quadratic Formula
The quadratic formula x = (-b ± √(b²-4ac)) / 2a solves any equation of the form ax² + bx + c = 0. It gives up to two solutions (roots) where the parabola crosses the x-axis. This is one of the most important formulas in algebra.
Understanding the Discriminant
The discriminant (b² - 4ac) determines the nature of the roots. If positive, there are two distinct real roots. If zero, there is exactly one real root (a repeated root). If negative, there are two complex conjugate roots and the parabola does not cross the x-axis.
Vertex and Axis of Symmetry
The vertex of the parabola is at x = -b/(2a), y = f(-b/(2a)). 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.
Applications of Quadratic Equations
Quadratic equations model projectile motion (height over time), area optimization, revenue/profit functions, braking distance, and many physics problems. The parabolic shape appears in satellite dishes, headlight reflectors, and suspension bridge cables.
Frequently Asked Questions
If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula cannot be applied. The single solution is x = -c/b.
When the discriminant is negative, the roots involve the imaginary unit i (√(-1)). For example, x² + 1 = 0 has roots x = i and x = -i. Complex roots always come in conjugate pairs (a+bi and a-bi).
Yes — first rearrange your equation into ax² + bx + c = 0 form by moving all terms to one side. Then identify the coefficients a, b, and c to enter into the calculator.
It is derived by completing the square on ax² + bx + c = 0. Divide by a, move c/a to the right, add (b/2a)² to both sides, factor the left as a perfect square, then take the square root and solve for x.