Problem 21 - Algebra 1: Concepts and Skills

Question

Find the discriminant of the quadratic equation. \(-2-x^{2}=4 x^{2}\)

Step-by-Step Solution

Answer

The discriminant of the quadratic equation \(-2 - x^2 = 4x^2\) is \(41\).

Step 1: Rewrite the quadratic equation in the standard form

Rearrange the given equation to the standard quadratic form: \(ax^2 + bx + c = 0\). So, \(-2-x^{2}=4 x^{2}\) becomes \(5x^2 + x - 2 = 0\), with \(a = 5\), \(b = 1\), and \(c = -2\).

Step 2: Identify the coefficients of the quadratic equation

Looking at the standard form of the equation, we can identify the coefficients \(a\), \(b\), and \(c\) as: \(a = 5\), \(b = 1\), and \(c = -2\). These will be substituted into discriminant formula.

Step 3: Find the discriminant

Using the discriminant formula \(D = b² - 4ac\), substitute the values gotten from Step 2 into the formula: \(D = (1)² - 4 * (5) * (-2)\). Therefore, the discriminant \(D\) is equal to \(41\).

Key Concepts

Standard Quadratic Form

When working with quadratic equations, it is essential to start with its most recognizable pattern, the standard quadratic form. This form is expressed as \(ax^2 + bx + c = 0\), where \(x\) represents the variable and \(a\), \(b\), and \(c\) are known as the coefficients of the equation. The value of \(a\) can't be zero, as this would make the equation linear, not quadratic.

Understanding the structure of this form is crucial since it sets the stage for further analysis of the equation, such as finding its roots or analyzing its graph. To transform an equation into this standard form, we typically rearrange terms and combine like terms. In the given exercise, the equation \(-2 - x^2 = 4x^2\) was reorganized to the standard form by moving all terms to one side, leading to \(5x^2 + x - 2 = 0\). This manipulation simplifies the identification of coefficients and subsequent steps in the problem-solving process.

Discriminant Formula

The discriminant of a quadratic equation is a powerful tool that provides insight into the nature of its roots without actually solving the equation. The discriminant formula, given as \(D = b^2 - 4ac\), determines the number and type of solutions. Here, \(a\), \(b\), and \(c\) are the coefficients from the standard quadratic form of the equation.

For example, if \(D > 0\), the quadratic equation has two distinct real roots. If \(D = 0\), it has one real root, and if \(D < 0\), the equation has complex roots. In the exercise, after identifying the coefficients from the standard form, we plug them into the discriminant formula to find \(D = (1)^2 - 4 * (5) * (-2) = 41\), indicating that our quadratic equation has two distinct real roots.

Quadratic Coefficients

The coefficients in a quadratic equation are the numerical factors that multiply each term represented by \(a\), \(b\), and \(c\) in the standard form \(ax^2 + bx + c = 0\). The coefficient \(a\) is tied to the square term and determines the curvature of the parabola when graphed. The coefficient \(b\) associates with the linear term, and \(c\) is the constant term.

The values of these coefficients play a critical role in the behavior of the quadratic function, including the direction in which the parabola opens, its width, and the location of its vertex. In our exercise, we first identified the coefficients as \(a = 5\), \(b = 1\), and \(c = -2\) after arranging the equation into the standard form. Understanding the significance of these coefficients can provide a deeper insight into the graphical representation and solutions of the equation.