Question 1

What is the quadratic formula?

Accepted Answer

For ax² + bx + c = 0, the roots are x = [−b ± √(b² − 4ac)] / 2a. The discriminant, b² − 4ac, tells you how many real roots exist: positive gives two, zero gives one repeated root, and negative gives two complex roots.

Question 2

What is the discriminant?

Accepted Answer

The discriminant is b² − 4ac. It determines the nature of the roots without solving fully. A positive discriminant means two distinct real roots, zero means one repeated real root, and a negative one means complex conjugate roots.

Question 3

How do I solve a quadratic by factoring?

Accepted Answer

Rearrange to zero and look for two numbers that multiply to ac and add to b, then split the middle term. For x² − 5x + 6 = 0, the numbers are −2 and −3, giving (x − 2)(x − 3) = 0, so x = 2 or x = 3.

Question 4

What if the coefficient a is zero?

Accepted Answer

If a = 0, the equation is not quadratic — it becomes linear, bx + c = 0, with the single solution x = −c/b. This solver expects a nonzero a for the quadratic formula.

Question 5

When do quadratic equations have no real solutions?

Accepted Answer

When the discriminant b² − 4ac is negative, the square root has no real value, so the equation has two complex roots. This happens when the parabola never crosses the x-axis — it is entirely above or below it.

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