Problem 67 - College Algebra Essentials

Question

Solve each equation in Exercises \(65-74\) using the quadratic formula. $$ x^{2}+5 x+3=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \[x = (-5 + \sqrt{13}) / 2\) and \(x = (-5 - \sqrt{13}) / 2\]

Step 1: Identify the coefficients

The coefficients for the equation \(x^{2}+5 x+3=0\) are a=1, b=5, c=3. We ought to substitute them into the quadratic formula.

Step 2: Substitute into the Quadratic Formula

Having identified the coefficients, we can substitute them into the quadratic formula: \[x = \[-b \pm \sqrt{b^{2}-4ac}\] / (2a) = \[-5 \pm \sqrt{(5)^{2}-4*(1)*(3)}\] / (2*1)\]

Step 3: Simplify the Result

Now we simplify the result: \[x = \[-5 \pm \sqrt{25-12}\] / 2 = \[-5 \pm \sqrt{13}\] / 2\]

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are polynomial equations of the second degree typically expressed in the form of \( ax^2 + bx + c = 0 \). To solve for \( x \) using the quadratic formula, one can simply plug in the values of the polynomial coefficients \( a \), \( b \), and \( c \) into the formula \( x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{{2a}} \).

This method provides a straightforward solution for finding the roots of any quadratic equation, even when factoring is not possible. The \( \pm \) symbol in the formula accounts for two possible solutions, since a quadratic equation can have two real roots, one real root (a repeated root), or no real roots (if the discriminant \( b^2-4ac \) is negative). Knowing how to apply the quadratic formula correctly is essential for any student dealing with quadratic equations.

Polynomial Coefficients

Polynomial coefficients are the numerical factors that precede the variable terms in a polynomial expression. In a quadratic equation of the form \( ax^2 + bx + c = 0 \), the coefficients are represented by \( a \), \( b \) and \( c \). Here, \( a \) is the coefficient of \( x^2 \) and is key to defining the equation as quadratic.

The coefficient \( b \) is linked to the \( x \) term, and \( c \) is the constant term, also known as the free term. These coefficients can be any real number, and they play a crucial role in determining the properties of the quadratic equation such as its graph's shape, orientation, and the position of its vertex. Understanding the significance of these coefficients is necessary to comprehend the behavior of quadratic functions and to correctly apply the quadratic formula in solving equations.

Equation Simplification

Simplifying an equation involves reducing it to a more basic form while preserving its solutions. When applying the quadratic formula, simplification is a vital step that can simplify the calculation process and avoid common errors. It includes combining like terms, eliminating unnecessary mathematical complexity, and performing arithmetic operations to make the equation as straightforward as possible.

For instance, after substituting the coefficients into the quadratic formula, you may obtain an expression under the square root (the discriminant) that can be evaluated, and a fraction that can be simplified. This step is crucial to arrive at the most reduced form of the solution, making it clearer and easier for interpretation. Simplification often involves radical expression reduction and rationalizing denominators, which are skills that improve with practice and strengthen one's understanding of algebra.