Quadratic Equation Solver (Discriminant & Vertex)

Q: What does the discriminant tell me?

For ax^2 + bx + c = 0 the discriminant D = b^2 - 4ac determines the nature of the roots: D > 0 gives two real roots, D = 0 gives one double real root, and D < 0 yields a pair of complex conjugate roots.

Q: How is the case a = 0 handled?

When a = 0 the equation becomes linear: bx + c = 0. If b ≠ 0 the solution is x = -c/b. If b = 0 and c = 0 there are infinitely many solutions, otherwise there is no solution.

How the Quadratic Formula Emerges

Start from ax² + bx + c = 0 and complete the square to obtain (x + b/2a)².
Take the square root of both sides to isolate x = [-b ± √(b² - 4ac)] / (2a).
The term under the radical is the discriminant D, whose sign dictates whether the roots are real or complex.

If a = 0 the expression is linear. The solver automatically switches to bx + c = 0 and reports the appropriate result.

Need step-by-step factorization, completing the square, or polynomial (≤3) walkthroughs? Try the quadratic & polynomial solver with steps.

FAQ

What does the discriminant tell me?

For ax² + bx + c = 0 the discriminant D = b² - 4ac determines the nature of the roots: D > 0 gives two real roots, D = 0 gives one double real root, and D < 0 yields a pair of complex conjugate roots.

How is the case a = 0 handled?

When a = 0 the equation becomes linear: bx + c = 0. If b ≠ 0 the solution is x = -c/b. If b = 0 and c = 0 there are infinitely many solutions; otherwise there is no solution.

How the Quadratic Formula Emerges

FAQ

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