Problem 76 - Algebra 2

Question

What is the discriminant of \(q x^{2}+r x+s=0 ?\) $$ \begin{array}{lllll}{\text { A. }|a+b|} & {\text { B. } q^{2}-4 r s} & {\text { C. } r^{2}-4 q s} & {\text { D. } s^{2}-4 q r}\end{array} $$

Step-by-Step Solution

Answer

The discriminant of the equation is C. r^2 - 4qs.

Step 1: Recall the discriminant formula for a quadratic equation

The discriminant of a quadratic equation in the form of ax^2 + bx + c = 0 is given by b^2 - 4ac.

Step 2: Identify the coefficients

In the equation qx^2 + rx + s = 0, identify q as 'a', r as 'b', and s as 'c' from the generic form of a quadratic equation.

Step 3: Apply the discriminant formula

Substitute the coefficients into the discriminant formula: discriminant = r^2 - 4 * q * s.

Key Concepts

Discriminant Formula

The discriminant is a valuable expression in algebra that reveals information about the roots of a quadratic equation. It's represented by the symbol \( \Delta \). The general formula to find the discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is \( \Delta = b^2 - 4ac \). This equation helps us determine the nature of the roots without actually solving the equation. If the discriminant is positive, there are two distinct real roots. If it is zero, there is exactly one real root (also called a repeated or double root). And finally, if the discriminant is negative, the quadratic equation has two complex roots.
Using the discriminant provides a quick way to analyze the quadratic equation's roots, which is essential for graphing parabolas or solving optimization problems. In the given exercise, the discriminant formula can be applied to the quadratic equation \(qx^2 + rx + s = 0\), where the coefficients correspond to those in the original discriminant formula.

Quadratic Equation

A quadratic equation is one of the most studied algebraic expressions in mathematics. It is generally written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are the coefficients, and \(x\) is the variable that needs to be solved for. The coefficient \(a\) must be non-zero, or else the equation becomes linear rather than quadratic.

Roots of a Quadratic Equation

A quadratic equation can have either two real roots, one real root, or two complex roots. These roots represent the \(x\)-values where the quadratic graph intersects the x-axis. The solutions or roots are found using various methods, including factoring, completing the square, using the quadratic formula, or graphing.
By understanding the structure and properties of a quadratic equation, students enhance their problem-solving skills, enabling them to tackle a wide range of mathematical challenges.

Coefficients of Quadratic Equation

In a quadratic equation of the form \(ax^2 + bx + c = 0\), the coefficients are the numerical parts that multiply each term of the equation. Coefficient \(a\) is linked to the term \(x^2\), coefficient \(b\) to the \(x\) term, and coefficient \(c\) is the constant term.

Role of Coefficients in Quadratics

Each coefficient has a specific role in shaping the graph of the quadratic equation. For instance:

• \(a\): Influences the direction (upward or downward) and the width of the parabola, which is the graph of the quadratic equation.
• \(b\): Affects the location of the vertex of the parabola along the x-axis.
• \(c\): Represents the y-intercept of the parabola or where it crosses the y-axis.

Understanding the significance of these coefficients is not only crucial for graphing the equation but also for determining the parabola's vertex and axis of symmetry. Through their manipulation, one can visually represent different real-world scenarios modeled by quadratic equations.