The Quadratic Formula and the Discriminant
The quadratic formula x = (-b ± √(b2 - 4ac)) / (2a) solves any equation ax2 + bx + c = 0. The discriminant b2 - 4ac, the part under the square root, tells you whether there are two, one, or zero real solutions before you even finish solving.
Factoring only works when a quadratic breaks into nice integer pieces. Most quadratics don't. The quadratic formula comes from completing the square on the general equation ax2 + bx + c = 0, so it rebuilds the solution from a, b and c no matter how ugly the numbers are. Plug in the three coefficients, respecting their signs, and the formula hands back every solution that exists, real or not.
The discriminant is the expression b2 - 4ac sitting under the radical. Its sign controls everything because you can't take the square root of a negative number and get a real result. If b2 - 4ac is positive, the radical splits into a plus branch and a minus branch, giving two distinct real solutions. If it's exactly zero, the plus and minus branches collapse to the same value, so there's exactly one real solution, a repeated root. If it's negative, the square root produces an imaginary number, so there are no real solutions at all, only a complex pair.
This connects directly to the parabola y = ax2 + bx + c. The solutions of ax2 + bx + c = 0 are exactly where that parabola crosses the x-axis. Two real solutions means two crossings; one real solution means the vertex just touches the axis; no real solutions means the parabola never reaches the axis at all, staying entirely above or below it. The discriminant is a shortcut: it reads the picture without ever drawing it or finishing the algebra.
Key terms
- discriminant
- the expression b2 - 4ac, whose sign reveals the number of real solutions a quadratic has.
- quadratic formula
- x = (-b ± √(b2 - 4ac)) / (2a), the formula that solves any equation ax2 + bx + c = 0.
- repeated root
- a single real solution that occurs when the discriminant equals zero, where the parabola's vertex touches the x-axis.
- complex solutions
- solutions involving the imaginary unit i that occur in a pair when the discriminant is negative.
Watch it explained
- How it worksHow does the discriminant under the radical decide between two, one and no real solutions?
- Worked exampleFind the solutions of 3x2 - 5x - 1 = 0 with the quadratic formula.
- Worked exampleFind how many real solutions 4x2 - 12x + 9 = 0 has, using the discriminant alone.
- Common mistakeWhy isn't a negative discriminant a sign that you slipped up somewhere in the arithmetic?
- Ties togetherHow does the discriminant tie a parabola's picture to the complex answers it is hiding?
Check yourself
End of topic.