key term - Distinct Roots

Definition

Distinct roots refer to the unique solutions or values of a quadratic equation that satisfy the equation. When a quadratic equation has distinct roots, it means the equation has two separate, non-overlapping solutions that can be graphed as two distinct points on the parabolic curve.

5 Must Know Facts For Your Next Test

1. Distinct roots occur when the discriminant $b^2 - 4ac$ is positive, meaning the equation has two real, non-equal solutions.
2. The number and nature of the roots of a quadratic equation can be determined by the sign of the discriminant: positive for distinct real roots, zero for one real root, and negative for complex conjugate roots.
3. Distinct roots can be found by solving the quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
4. Graphically, distinct roots are represented as two separate points on the parabolic curve of the quadratic function.
5. The presence of distinct roots affects the shape and properties of the graph, such as the location of the vertex and the direction of the parabola.

Review Questions

• Explain how the discriminant of a quadratic equation is used to determine the nature of the roots.
  • The discriminant of a quadratic equation, given by the expression $b^2 - 4ac$, is used to determine the nature of the roots. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has two complex conjugate roots. The sign of the discriminant is a crucial factor in understanding the behavior of the quadratic equation and its graphical representation.
• Describe the relationship between distinct roots and the graph of a quadratic function.
  • When a quadratic equation has distinct roots, the graph of the corresponding quadratic function will have two separate points where the parabola intersects the x-axis. These points represent the distinct solutions to the equation. The presence of distinct roots affects the shape and properties of the graph, such as the location of the vertex and the direction of the parabola. Graphically, distinct roots are represented as two separate points on the parabolic curve, which can provide valuable insights into the behavior and characteristics of the quadratic function.
• Analyze how the quadratic formula can be used to find distinct roots and explain the significance of this process.
  • The quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, can be used to find the distinct roots of a quadratic equation. By plugging in the values of $a$, $b$, and $c$ into the formula and calculating the two solutions, you can determine the distinct roots of the equation. The significance of this process lies in its ability to provide the exact values of the roots, which are crucial for understanding the behavior of the quadratic function and its graphical representation. Knowing the distinct roots allows you to accurately sketch the parabolic curve, locate the points of intersection with the x-axis, and make informed decisions about the properties and characteristics of the quadratic function.

Related terms

Quadratic Equation:

A polynomial equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$.

Parabola:

The graph of a quadratic function, which is a U-shaped curve.

Discriminant:

The value $b^2 - 4ac$ that determines the nature of the roots of a quadratic equation.