# Is it possible to have different quadratic equations with the same solution? Explain.

## Question:

Is it possible to have different quadratic equations with the same solution? Explain.

The number, as well as the types of solutions of a specific quadratic equation both depend on the value of the discriminant. Discriminant defines whether a quadratic function (parabola) intersects the x-axis. If parabola intersects the x-axis at its vertex, we will have only one distinct solution of such a quadratic equation with a multiplicity of 2 (that is, two solutions having the same value).

Assume that a quadratic equation is given by:

{eq}ax^2 + bx + c = 0 {/eq}

We already know that such a quadratic equation will have one solution if the parabola {eq}f(x) = ax^2 + bx + c {/eq} intersects the x-axis at its vertex. Algebraically speaking, the discriminant, defined by {eq}D = b^2 - 4ac {/eq}, should have a value of 0:

{eq}D = b^2 - 4ac = 0\\ b^2 - 4ac = 0 {/eq}

Let's find the relationship between the coefficients to see when this relationship holds:

{eq}b^2 = 4ac\\ b = \pm 2\sqrt{ac} {/eq}

Thus, if this relationship holds between coefficients a, b and c, our quadratic equation will have one distinct solution.