Problem 28 - Precalculus : Building Concepts and Connections

Question

Solve the inequality by factoring. $$x^{2}+4 x+4>0$$

Step-by-Step Solution

Answer

The solution to the inequality \(x^2 + 4x + 4 > 0\) is \(x \in (-\infty, -2) \cup (-2, \infty)\).

Step 1: Factorizing the Quadratic Expression

The given inequality is \(x^{2}+4x+4 > 0\). Since the coefficients of this quadratic equation are perfect squares, we can factorize this equation as follows: \(x^2 + 4x + 4 = (x + 2)^2\). So the inequality becomes \((x+2)^2 > 0\).

Step 2: Solve the Factored Inequality

To solve this inequality, it's helpful to consider when is square of a number greater than zero. Clearly, except for \(x=-2\), square of a number is always positive for all real values of x. It means the solution to this inequality is \(x \in (-\infty, -2) \cup (-2, \infty)\). The set \(-2\) is excluded because square of any number cannot be negative.

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions is a fundamental technique in algebra that simplifies complex equations and makes them easier to solve. It involves breaking down a quadratic equation into a product of simpler expressions, typically binomials. For example, the quadratic expression \(x^2 + 4x + 4\) can be factored as \(x + 2)^2\), where \(x + 2\) is a binomial. This reveals that \(x + 2\) is repeated, which is indicative of a 'perfect square trinomial'—a special form where the first and last terms are squares of binomials and the middle term is twice the product of the binomials.

Understanding the structure of perfect squares is valuable as they have predictable patterns. When factoring, always look for a common factor first. If there isn't one, you can use techniques such as the 'difference of squares', 'sum/difference of cubes', or for more complex expressions, the quadratic formula. Once factored, an equation becomes simpler to evaluate, especially when used within other algebraic processes such as solving inequalities or graphing functions.

Inequality Solutions

An inequality solution represents a range of values that satisfy the inequality equation. Unlike a quadratic equation that typically has a fixed number of solutions, an inequality often has a continuous range of solutions or an interval. In the given inequality \(x^2 + 4x + 4 > 0\), after factoring, we are tasked to determine when the square of a binomial \(x + 2\) is greater than zero.

Inequality solutions can be visualized on a number line, where one can plot the critical points that satisfy the equality part (in this case, \(x = -2\)) and then determine which intervals on the line satisfy the inequality. Since the inequality is strict \(>\) and not \(\geq\), the value of \(x = -2\) is not part of the solution set, and therefore we exclude it, representing the solution as an open interval on the number line. The solution set is all values of \(x\) except \(x = -2\), indicating that \(x\) can be any real number other than \(x = -2\). Inequality solutions are often depicted with parenthesis or brackets depending on if the endpoint values are included (closed interval) or not (open interval).

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not zero. The solutions to these equations are found using a variety of methods such as factoring, completing the square, or applying the quadratic formula. The roots of a quadratic equation are the \(x\)-values that make the equation equal to zero, and they can be real or complex numbers.

The nature of the roots is determined by the discriminant \(\Delta = b^2 - 4ac\). If \(\Delta > 0\), there are two distinct real solutions. If \(\Delta = 0\), there is exactly one real solution, which is also known as a repeated or double root. Lastly, if \(\Delta < 0\), there are two complex solutions. The quadratic equation plays a key role in various areas of mathematics and science, and learning to solve them is essential for advancing in these subjects. Each method of solving has its own application and they are often used in conjunction with each other to solve real-world problems.