Quadratic expression

Definition

A quadratic expression is a polynomial expression of degree two, typically written in the form $$ax^2 + bx + c$$, where 'a', 'b', and 'c' are constants, and 'x' is the variable. This type of expression represents a parabola when graphed and can have various applications in solving real-world problems, such as finding areas or modeling projectile motion.

5 Must Know Facts For Your Next Test

1. The standard form of a quadratic expression is $$ax^2 + bx + c$$, where 'a' cannot be zero; otherwise, it becomes a linear expression.
2. Quadratic expressions can be expressed in vertex form as $$a(x-h)^2 + k$$, where (h,k) represents the vertex of the parabola.
3. The graph of a quadratic expression is always a parabola that opens either upward or downward depending on the sign of 'a'.
4. The axis of symmetry for the graph of a quadratic expression can be found using the formula $$x = -\frac{b}{2a}$$.
5. Quadratic expressions can have zero, one, or two real roots, which can be determined using the discriminant value $$b^2 - 4ac$$.

Review Questions

• How do you identify the characteristics of a quadratic expression based on its coefficients?
  • The coefficients 'a', 'b', and 'c' in a quadratic expression $$ax^2 + bx + c$$ determine key features like the direction in which the parabola opens and its vertex location. If 'a' is positive, the parabola opens upward; if negative, it opens downward. The value of 'b' affects the placement of the vertex along the x-axis, while 'c' represents the y-intercept. Understanding these relationships helps in graphing and analyzing the quadratic function.
• Explain how factoring a quadratic expression can lead to finding its roots and why this process is important.
  • Factoring a quadratic expression allows us to rewrite it as a product of two binomials, such as $$(x - r_1)(x - r_2)$$, where $$r_1$$ and $$r_2$$ are its roots. This approach is crucial because it simplifies finding the values of 'x' that make the expression equal to zero, which are essential in solving quadratic equations. By setting each factor to zero, we can easily identify all possible solutions to the equation.
• Analyze how the discriminant impacts the solutions of a quadratic expression and what it reveals about its graph.
  • The discriminant, given by $$D = b^2 - 4ac$$ for a quadratic expression $$ax^2 + bx + c$$, plays a critical role in determining the nature of its roots. If D is positive, there are two distinct real roots, indicating that the parabola crosses the x-axis at two points. If D equals zero, there is exactly one real root, meaning the parabola touches the x-axis at its vertex. When D is negative, there are no real roots; instead, the parabola remains entirely above or below the x-axis. This information provides insight into how the quadratic function behaves visually on a graph.

Related terms

Polynomial: An algebraic expression that consists of terms combined using addition, subtraction, and multiplication, where each term includes a variable raised to a non-negative integer exponent.

Factoring: The process of breaking down a quadratic expression into products of simpler expressions, which can help in solving equations or simplifying calculations.

Discriminant: A value calculated from the coefficients of a quadratic expression that determines the nature of its roots, helping to identify if they are real or complex.