Problem 65 - College Algebra Essentials

Question

Solve each equation in Exercises \(65-74\) using the quadratic formula. $$ x^{2}+8 x+15=0 $$

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2}+8x+15=0\) are \(x=-5\) and \(x=-3\)

Step 1: Identify the Coefficients

In our equation \(x^{2} + 8x + 15 = 0\), the coefficients a, b, and c are 1, 8 and 15, respectively.

Step 2: Substitute into the Quadratic Formula

We substitute these values into the quadratic equation: \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\), we get: \(x=\frac{-8\pm\sqrt{8^{2}-4(1)(15)}}{2(1)}\).

Step 3: Evaluate the Expression

Calculate the expression under the square root first: \(x=\frac{-8\pm\sqrt{64-60}}{2}\), which simplifies to \(x=\frac{-8\pm\sqrt{4}}{2}\).

Step 4: Final Calculation

Evaluate the square root and perform the addition and subtraction separately to find two possible solutions: \(x=\frac{-8\pm2}{2}\), which gives us \(x=-5\) or \(x=-3\).

Key Concepts

Solving Quadratic Equations

Quadratic equations can seem daunting at first, but they follow a predictable pattern that, once understood, can be easily solved—especially with the quadratic formula. This formula gives a clear method for finding the x-values that satisfy the quadratic equation of the form \( ax^2 + bx + c = 0 \). The steps are straightforward: identify the coefficients, substitute them into the formula, evaluate the resulting expression, and then simplify to find the solutions.
What makes the quadratic formula so versatile is its ability to handle any coefficients and provide solutions that may be real or complex numbers. Regardless of whether the solutions are whole numbers, fractions, or involve square roots, the quadratic formula is a reliable tool to find them.

Quadratic Equation Coefficients

The coefficients of a quadratic equation—typically denoted as \(a\text{, }b\text{, and }c\)—play a crucial role in its behavior and solution. In the equation \(x^2 + 8x + 15 = 0\), the coefficient \(a\) represents the quadratic term (1 in this case), \(b\) is the linear coefficient (8), and \(c\) stands for the constant term (15).

Different values for these coefficients affect the parabola's direction, width, and where it intersects the x-axis. For instance, when \(a\) changes sign, the parabola flips direction. When the magnitude of \(a\) grows, the parabola becomes steeper. These coefficients are plugged into the quadratic formula and determine the solutions that are found. Understanding their role can help visualize the equation’s graph and predict the nature of its solutions.

Evaluating Expressions

Evaluating expressions is a fundamental skill in algebra that involves performing operations in a specific order, often guided by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). When using the quadratic formula, it's essential to carefully evaluate the expression under the square root (the discriminant) first, as this determines the number and type of solutions. For example, in \(x=\frac{-8\pm\sqrt{64-60}}{2}\), we focus on calculating the square root of \(\sqrt{64-60}\) before anything else.

Once this is done, further simplification includes adding and subtracting terms and finally dividing by the coefficient of the quadratic term. This step-by-step simplification ultimately yields the roots of the quadratic equation—values that are critical in understanding the equation's solutions in both numerical and graphical contexts.