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Quadratic Roots and Discriminant

Quadratic Equations · Class X

Tune a, b, c and see the nature of roots flip between real/distinct, repeated, or complex.

1
type a value
-55
-5
type a value
-1010
6
type a value
-1010
Live values
  • Discriminant1
  • Root 13
  • Root 22
  • Sum of roots5
  • Product of roots6
xy
  • y = a x^2 + b x + c

Formulas in this lab

  • Discriminant
    Δ=b2−4ac\Delta = b^2 - 4ac
  • Root 1
    −b+Δ2a\frac{-b + \sqrt{\Delta}}{2a}
  • Root 2
    −b−Δ2a\frac{-b - \sqrt{\Delta}}{2a}
  • Sum of roots
    −b/a-b/a
  • Product of roots
    c/ac/a
Tip: Discriminant = 0 means the parabola just grazes the x-axis , one repeated root.

Frequently asked questions

▶How does the discriminant tell us about the roots?

The discriminant is $\Delta = b^2 - 4ac$. If $\Delta > 0$ the equation has two distinct real roots, if $\Delta = 0$ there is one repeated root, and if $\Delta < 0$ the roots are complex conjugates. The lab marks each region in a different colour so you can see the boundary.

▶How do I use this lab?

Slide a, b, c. The lab shows the discriminant, both roots, the vertex and the parabola. Try b = 0 to see a symmetric U; try c = 0 to see one root pinned at the origin.

▶Common mistake on this topic

Many students forget the minus sign in $-\frac{b}{2a}$ when finding the vertex, or forget the leading 2 in the denominator. Always write the formula before plugging in numbers.

▶Where do we see this in real life?

Quadratics describe a ball thrown up, the shape of a satellite dish, and the profit-vs-price curve for a small shop owner. The roots tell you when the ball lands or where break-even happens.