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.
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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.