The Discriminant and Nature of Roots

The discriminant is the expression under the square root in the quadratic formula: [formula] It tells you how many real solutions the equation ax^2 + bx + c = 0 has, without solving: | Discriminant | Number of real solutions | Graph | |---|---|---| | Delta > 0 | Two distinct real solutions | Parabola crosses x-axis twice | | Delta = 0 | One repeated real solution | Parabola touches x-axis at v…

Why it works: in x = frac{-b pm sqrt{Delta}}{2a}: - If Delta > 0: sqrt{Delta} is a real positive number → two different values from pm - If Delta = 0: sqrt{0} = 0 → pm 0 gives only one value - If Delta < 0: sqrt{text{negative}} is not real → no real solutions Bonus: if Delta is a perfect square (1, 4, 9, 16, ...), the solutions are rational (nice fractions). If not, they contain square roots.

Example: Without solving, how many real solutions does x^2 + 3x + 5 = 0 have?