The Quadratic Formula Explained Simply (With Examples)
ax² + bx + c = 0 looks intimidating. The quadratic formula solves it every time, and it is easier than you think.
The quadratic formula: x = (-b ± sqrt(b²-4ac)) / 2a. It solves any equation in the form ax² + bx + c = 0. You plug in a, b, and c, do the arithmetic, and get two answers (the ± gives you both roots). That is it — one formula handles every quadratic equation in existence.
Example: x² - 5x + 6 = 0
Here a=1, b=-5, c=6. Discriminant: (-5)²-4(1)(6) = 25-24 = 1. Square root of 1 is 1. x = (5 ± 1) / 2. So x = 6/2 = 3 or x = 4/2 = 2. The solutions are x=2 and x=3. You can verify: 2²-5(2)+6 = 4-10+6 = 0. Correct.
What the Discriminant Tells You
The discriminant (b²-4ac) reveals the nature of the solutions before you solve. Positive discriminant: two different real solutions (the parabola crosses the x-axis twice). Zero discriminant: one repeated solution (the parabola touches the x-axis at exactly one point). Negative discriminant: two complex solutions with imaginary parts (the parabola never touches the x-axis). Knowing this instantly tells you what type of answer to expect.