For a quadratic function $ax^2 + bx + c = 0$, roots can be found using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
The Discriminant $\Delta = b^2 - 4ac$ determines the nature of roots: if $\Delta > 0$, there are two distinct real roots; if $\Delta = 0$, there is one real root; if $\Delta < 0$, there are two complex conjugate roots.
Roots can also be identified graphically as the points where a parabola intersects the x-axis.
The sum of the roots of a quadratic equation $ax^2 + bx + c = 0$ is given by $-\frac{b}{a}$ and their product is $\frac{c}{a}$.
Factoring is another method to find roots, where you express the quadratic equation in factored form $(x - r_1)(x - r_2) = 0$ and solve for $r_1$ and $r_2$.
Review Questions
How do you determine the number and type of roots for a quadratic equation?
What does it mean when we say that a number is a root of a polynomial function?
Using the quadratic formula, find the roots of the equation $3x^2 - 4x + 1 = 0$.
Related terms
Quadratic Formula: A formula that provides solutions to quadratic equations: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Discriminant: $\Delta = b^2 - 4ac$, used to determine the nature and number of solutions for a quadratic equation.
Vertex: The highest or lowest point on the graph of a parabola, given by $(h,k)$ where $h=-\frac{b}{2a}$ and $k=f(h)$.