Problem 41 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation using any convenient method. $$ x^{2}+3 x+1=0 $$

Step-by-Step Solution

Answer

The solutions for the equation \(x^2 + 3x + 1 = 0\) are \(x = [-3 + \sqrt{5}] / 2\) and \(x = [-3 - \sqrt{5}] / 2\).

Step 1: Identification of coefficients

For the equation \(x^2 + 3x + 1 = 0\), the coefficients are a=1, b=3 and c=1.

Step 2: Apply the quadratic formula

Put the coefficients a, b, and c into the quadratic formula: \(x = [-b ± sqrt(b^2 - 4ac)] / 2a\). Hence, we get: \(x = [-3 ± sqrt(3^2 - 4*1*1)] / 2*1\).

Step 3: Simplify the expression

Simplify the above expression to solve for x. Hence, the solution to the equation will be \(x = [-3 ± sqrt(9 - 4)] / 2 = [-3 ± sqrt(5)] / 2\). Therefore, the two solutions are \(x = [-3 + sqrt(5)] / 2\) and \(x = [-3 - sqrt(5)] / 2\).

Key Concepts

Solving Quadratic Equations

A quadratic equation is a second-degree polynomial typically written in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \), are coefficients and \( a \) is not equal to zero. The process to solve such equations involves finding the values of \( x \) which satisfy the equation. There are several methods available for solving quadratic equations including factoring, completing the square, using the quadratic formula, and graphing.

For many students, using the quadratic formula is a reliable and straightforward method, especially when the equation does not factor easily. This method can be applied to any quadratic equation and provides a systematic approach to finding the roots. When solving an equation, one must first ensure that it is in standard form, then identify the coefficients corresponding to \( a \), \( b \), and \( c \) before substituting them into the quadratic formula.

Quadratic Formula

The quadratic formula provides a solution for the roots of any quadratic equation and is expressed as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, the term \( \sqrt{b^2 - 4ac} \) inside the formula is called the discriminant. The value of the discriminant can indicate the nature of the roots. If it's positive, the equation has two real and distinct solutions. If it's zero, the equation has exactly one real solution (repeated root). If it's negative, the equation has two complex solutions.

To use the quadratic formula, one extracts the coefficients \( a \) (coefficient of \( x^2 \)), \( b \) (coefficient of \( x \)), and \( c \) (constant term) from the equation and plugs them into the formula. This direct substitution makes it an effective method, especially when other methods like factoring are not feasible. Students must remember to perform the operations within the formula carefully, paying particular attention to the signs plus (\( + \)) and minus (\( - \)) and the order of arithmetic operations.

Coefficients of a Quadratic Equation

The coefficients of a quadratic equation refer to the numerical factors of the terms in the standard form \( ax^2 + bx + c = 0 \). The coefficient \( a \) is associated with the \( x^2 \) term and is crucial as it determines the equation's curvature and direction of its parabolic graph. Coefficient \( b \) is linked to the \( x \) term and influences the axis of symmetry of the parabola. Coefficient \( c \) represents the constant term and affects the y-intercept of the graph.

Understanding these coefficients is key as they not only impact the graph's shape and position but also play a vital role in determining the nature and number of the roots of the equation through the discriminant \( (b^2 - 4ac) \). The coefficients must be accurately identified and used in the quadratic formula to ensure the correct solutions are obtained.