What does the discriminant tell me?
Positive means two distinct real roots, zero means one repeated root, negative means two complex roots. It answers the shape question before you solve.
Solve any quadratic equation using the quadratic formula. Find real and complex roots, discriminant, vertex, and axis of symmetry.
For ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ÷ 2a. Discriminant D = b² − 4ac. Vertex at x = −b ÷ 2a.
For 2x² + 5x − 3 = 0 the discriminant is 25 + 24 = 49, so x = (−5 ± 7) ÷ 4, giving roots of 0.5 and −3.
Positive means two distinct real roots, zero means one repeated root, negative means two complex roots. It answers the shape question before you solve.
Factor when the coefficients are small integers and a factor pair is obvious. The quadratic formula always works, so use it whenever factoring stalls.
Use x = −b ÷ 2a, then substitute back for y. It is the minimum when a is positive and the maximum when a is negative.