Roots of a polynomial are the values of the variable that make the polynomial equal to zero. They are also known as solutions or zeros of the equation.
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The roots of a quadratic function can be found using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
A quadratic equation may have two real roots, one real root (a repeated root), or two complex roots depending on the discriminant ($b^2 - 4ac$).
If the discriminant is positive, there are two distinct real roots; if it is zero, there is exactly one real root (repeated); and if it is negative, there are no real roots but two complex conjugate roots.
The sum of the roots of a quadratic equation $ax^2 + bx + c = 0$ is given by $-\frac{b}{a}$, and their product is given by $\frac{c}{a}$.
Graphically, roots correspond to the points where the graph of the polynomial intersects the x-axis.
Review Questions
What does it mean for a value to be a root of a polynomial?
How can you determine whether a quadratic equation has real or complex roots?
What is the relationship between the coefficients of a quadratic equation and its roots?