Find a value of c so that y = x^2 - 10x + c has exactly one x-intercept.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form y = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of 'a'. Understanding the properties of quadratic functions is essential for analyzing their intercepts and vertex.

X-Intercepts

X-intercepts are the points where a graph crosses the x-axis, which occur when the output value (y) is zero. For a quadratic function, the x-intercepts can be found by solving the equation ax^2 + bx + c = 0. The number of x-intercepts is determined by the discriminant (b^2 - 4ac) of the quadratic equation.

Discriminant

The discriminant is a key component of the quadratic formula, given by the expression b^2 - 4ac. It determines the nature of the roots of the quadratic equation: if the discriminant is positive, there are two distinct real roots; if it is zero, there is exactly one real root (a repeated root); and if negative, there are no real roots. For the quadratic to have exactly one x-intercept, the discriminant must equal zero.