Problem 7 - Precalculus : Building Concepts and Connections

Question

Use the quadratic formula to solve the equation. $$x^{2}-3 x-2=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \frac{3 + \sqrt{17}}{2}\) and \(x = \frac{3 - \sqrt{17}}{2}\)

Step 1: Identifying Coefficients

Identify the coefficients from the original quadratic equation. In this case, a is 1, b is -3 and c is -2.

Step 2: Plugging into the Quadratic Formula

Now substitute a, b and c into the quadratics formula which is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Resulting equation will be \(x = \frac{3 \pm \sqrt{(-3)^2 - 4 * 1 * -2}}{2 * 1}\).

Step 3: Solving the Equation

Simplify the equation. First inside the square root: 9 - (-8) = 17. Substituting back into the equation, we get \(x = \frac{3 \pm \sqrt{17}}{2}\).

Key Concepts

Solving Quadratic Equations

Quadratic equations can be found throughout algebra and introduce students to concepts that apply in advanced mathematics, physics, engineering, and many more fields. The general form of a quadratic equation is written as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients and \(x\) represents the variable.
To solve these equations, we can use a variety of methods, but one of the most reliable is the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The formula is an exceptional tool because it gives us the ability to find solutions for both real and complex numbers, catering to a broad range of quadratic equations.
When applying the quadratic formula, it is crucial to first identify the values of \(a\), \(b\), and \(c\) from the equation. Once these coefficients are determined, they are substituted into the formula, which often involves simplifying under the square root (the discriminant) before finding the final solutions for \(x\).

Coefficient Identification

In algebraic expressions, the coefficients are the numerical factors that accompany the variables. Identifying coefficients correctly is a key step in solving equations, particularly when using the quadratic formula. For example, in the quadratic equation \(x^2 - 3x - 2 = 0\), the coefficient \(a\) is associated with \(x^2\) and is 1 (since no number before \(x^2\) implies it's 1), \(b\) is -3, and \(c\) is -2.
It is important to note that the signs of the coefficients affect the calculations significantly. A positive or negative sign can change the outcome when you plug these values into the formula, especially within the square root. Careful attention must be paid to these details to avoid common mistakes and ensure accurate solutions.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and arithmetic operators. In the context of quadratic equations, the expressions become slightly more complex, including squares of variables and their products with coefficients. Understanding how to manipulate these expressions is fundamental to finding solutions to quadratic equations.
Taking the example given, \(x^2 - 3x - 2 = 0\), each term in this expression represents a specific component that contributes to the structure of the equation. The first term \(x^2\) is the squared term, the second term \(3x\) is the linear term, and the constant \(2\) is the term without a variable. Manipulation of these expressions ranges from basic arithmetic to factoring and expanding binomials. Mastery of working with algebraic expressions will provide students the foundation to tackle not just quadratic equations, but a myriad of algebraic challenges.