$ax^2 + bx + c$

Definition

$ax^2 + bx + c$ is a general quadratic equation, where $a$, $b$, and $c$ are constants. This type of equation is fundamental in the study of quadratic functions and their applications in various mathematical and scientific fields.

5 Must Know Facts For Your Next Test

1. The constant $a$ in the equation $ax^2 + bx + c$ determines the shape of the parabolic graph, with $a > 0$ resulting in an upward-opening parabola and $a < 0$ resulting in a downward-opening parabola.
2. The constant $b$ in the equation $ax^2 + bx + c$ determines the horizontal shift of the parabola, with the vertex located at $(-b/(2a), f(-b/(2a)))$.
3. The constant $c$ in the equation $ax^2 + bx + c$ determines the vertical shift of the parabola, with the $y$-intercept located at $(0, c)$.
4. Quadratic equations can be solved using various methods, including factoring, completing the square, and using the quadratic formula.
5. Quadratic equations have either two real roots, one real root (a repeated root), or no real roots, depending on the value of the discriminant $b^2 - 4ac$.

Review Questions

• Explain how the values of $a$, $b$, and $c$ in the equation $ax^2 + bx + c$ affect the shape and position of the parabolic graph.
  • The value of $a$ determines the shape of the parabola, with $a > 0$ resulting in an upward-opening parabola and $a < 0$ resulting in a downward-opening parabola. The value of $b$ determines the horizontal shift of the parabola, with the vertex located at $(-b/(2a), f(-b/(2a)))$. The value of $c$ determines the vertical shift of the parabola, with the $y$-intercept located at $(0, c)$. Together, the values of $a$, $b$, and $c$ define the overall shape, position, and characteristics of the quadratic function.
• Describe the different methods that can be used to solve quadratic equations of the form $ax^2 + bx + c = 0$.
  • Quadratic equations of the form $ax^2 + bx + c = 0$ can be solved using various methods, including factoring, completing the square, and using the quadratic formula. Factoring involves finding two factors of the equation that, when multiplied, give the original equation. Completing the square involves manipulating the equation to obtain a perfect square, which can then be solved. The quadratic formula, given by $x = (-b \pm \sqrt{b^2 - 4ac})/(2a)$, can be used to directly calculate the roots of the equation. The choice of method often depends on the specific values of $a$, $b$, and $c$, and the desired level of accuracy and efficiency in the solution.
• Analyze the relationship between the discriminant $b^2 - 4ac$ and the number and nature of the real roots of a quadratic equation $ax^2 + bx + c = 0$.
  • The discriminant $b^2 - 4ac$ plays a crucial role in determining the number and nature of the real roots of a quadratic equation $ax^2 + bx + c = 0$. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root (a repeated root). If the discriminant is negative, the equation has no real roots, and the roots are complex conjugates. The sign and magnitude of the discriminant provide important information about the behavior of the quadratic function and its applications, such as the existence and nature of solutions, the symmetry of the parabolic graph, and the optimization of related problems.

Related terms

Quadratic Function: A quadratic function is a polynomial function of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers, and $a \neq 0.

Vertex: The vertex of a quadratic function $f(x) = ax^2 + bx + c$ is the point on the graph where the function changes direction, and it is given by the coordinates $(-b/(2a), f(-b/(2a)))$.

Discriminant: The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is the expression $b^2 - 4ac$, which determines the number and nature of the real roots of the equation.