The Discriminant and the Number of Real Solutions
The discriminant is the expression b2-4ac hiding inside the quadratic formula. Its sign alone tells you whether ax2+bx+c=0 has two real solutions, exactly one, or none.
A quadratic's graph is a parabola, and how many real solutions its equation has just means how many times that parabola crosses the x-axis: twice, once (grazing it at the vertex), or never. Sliding the parabola up or down shifts the vertex's height without changing its shape, so that slide is exactly what pushes it from crossing twice, to touching once, to missing the axis completely.
The quadratic formula is x = (-b +- √(b2-4ac)) / 2a. Everything about the count of real solutions lives inside that square root. If b2-4ac is positive, the sqrt gives two different real numbers, so plus and minus produce two distinct roots. If b2-4ac equals zero, the sqrt is zero, so plus and minus give the same single answer: one real solution, a repeated root at the vertex. If b2-4ac is negative, you'd need the square root of a negative number, which isn't real, so there are no real solutions at all. For 3x2-6x+4=0, the discriminant is (-6)2-4(3)(4) = 36-48 = -12, negative, so no real solutions, without ever finishing the computation. For x2+kx+9=0 to have exactly one real solution, set k2-4(1)(9)=0, so k2=36 and k=6 or k=-6.
This connects directly to graphing parabolas: the discriminant's sign predicts intercepts before you plot anything, linking the algebra of the equation to the picture of the graph. It also sets up later work with complex numbers, since a negative discriminant doesn't mean the solutions disappear, it means they move off the real number line entirely.
Key terms
- Discriminant
- The expression b2-4ac from a quadratic's coefficients, whose sign reveals the number of real solutions.
- Repeated root
- The single real solution that occurs when the discriminant is zero, located exactly at the parabola's vertex.
- x-intercept
- A point where a graph crosses the x-axis, corresponding to a real solution of the equation set equal to zero.
- Quadratic formula
- The formula x = (-b +- √(b2-4ac)) / 2a that solves any equation of the form ax2+bx+c=0.
Watch it explained
- How it worksHow does b2-4ac under the radical decide between two real answers, one, and none at all?
- Worked exampleFind how many real solutions 3x2-6x+4=0 has, without actually solving it.
- Worked exampleFind every value of k that makes x2+kx+9=0 have exactly one real solution.
- Common mistakeWhy doesn't a quadratic with no real solutions mean the parabola has vanished from the plane?
- Going furtherWhat happens to the two roots as the discriminant shrinks down toward zero?
Check yourself
End of topic.