Problem 66 - College Algebra Essentials

Question

Solve each equation in Exercises \(65-74\) using the quadratic formula. $$ x^{2}+8 x+12=0 $$

Step-by-Step Solution

Answer

The solutions for the equation are \(x = -2\) and \(x = -6\).

Step 1: Identifying a, b and c

First, the coefficients in the quadratic equation are identified as \(a = 1\), \(b = 8\), \(c = 12\).

Step 2: Plugging a, b and c into the quadratic formula

The values of \(a\), \(b\) and \(c\) are then substituted into the quadratic formula \((-b ± √(b^2 - 4ac)) / (2a)\), resulting in \((-8 ± √((8)^2 - 4*1*12)) / (2*1)\).

Step 3: Solving the equation

After simplifying the equation, the two possible solutions for \(x\) are found to be \((-8 + √(64 - 48)) / 2\) and \((-8 - √(64 - 48)) / 2\), which further simplify to \((-8 + √16) / 2\) and \((-8 - √16) / 2\). This gives us \(x = -2\) and \(x = -6\).

Key Concepts

Solving Quadratic Equations

Quadratic equations form the bedrock of algebra and can often be encountered in various mathematical and real-world applications. They are characterized by an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are known values, and \(a \eq 0\). The task of solving these equations may seem daunting at first, but by employing the quadratic formula, you unlock a straightforward method to find the roots.

Solving quadratic equations requires you to identify the coefficients of \(x^2\), \(x\), and the constant term, which are \(a\), \(b\), and \(c\) respectively. Each coefficient plays a critical role in determining the curve's characteristics, such as its direction (upward or downward), its width, and the location of its vertex. The quadratic formula provides a way to find the values of \(x\) that make the equation true, known as 'solutions' or 'roots'.

Quadratic Equation Coefficients

In the context of quadratic equations like \(x^2 + 8x + 12 = 0\), the coefficients are the numerical parts of the terms. Here, \(a = 1\) is the coefficient before \(x^2\), \(b = 8\) is the coefficient before \(x\), and \(c = 12\) is the constant term. These coefficients determine the shape and position of the parabolic graph of the equation. The coefficient \(a\) affects the opening direction and the width of the parabola. When \(a > 0\), the parabola opens upwards; when \(a < 0\), it opens downwards. The larger the absolute value of \(a\), the narrower the parabola.

The coefficient \(b\) impacts the axis of symmetry of the parabola, which is the line that divides it into two mirror-image halves. The constant \(c\) represents the \(y\)-intercept of the graph, the point where the parabola crosses the \(y\)-axis. Understanding these coefficients helps in visualizing the graph and provides insights that can be instrumental in solving quadratic equations more intuitively.

Quadratic Formula Steps

The quadratic formula, \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is a powerful tool for finding the roots of a quadratic equation. To apply this formula, follow these systematic steps:

• Identify the coefficients: Determine the values of \(a\), \(b\), and \(c\) from the quadratic equation \(ax^2 + bx + c = 0\).
• Substitute coefficients into the formula: Plug the values of \(a\), \(b\), and \(c\) into the quadratic formula.
• Calculate the discriminant: Compute the value of the discriminant \(b^2 - 4ac\). This step is crucial as it indicates the nature and number of solutions.
• Find the two solutions: Apply the plus-minus symbol (\(\pm\)) to find both possible solutions for \(x\).
• Simplify your results: Simplify the expressions to find the most straightforward form of both solutions.

Once you apply these steps, you'll end up with either two real solutions, one real solution, or two complex solutions, depending on the discriminant's value.