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Roots

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Algebra and Trigonometry

Definition

Roots are the values of $x$ that satisfy the equation $f(x)=0$. In the context of polynomial functions, they are also known as zeros or solutions.

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5 Must Know Facts For Your Next Test

  1. For a quadratic function $ax^2 + bx + c = 0$, roots can be found using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
  2. 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.
  3. Roots can also be identified graphically as the points where a parabola intersects the x-axis.
  4. The sum of the roots of a quadratic equation $ax^2 + bx + c = 0$ is given by $-\frac{b}{a}$ and their product is $\frac{c}{a}$.
  5. 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 $3x^2 - 4x + 1 = 0$.
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