Problem 33 - Algebra for College Students

Question

Solve each inequality. $$4 x^{2}+20 x+25 \leq 0$$

Step-by-Step Solution

Answer

The solution is \(x = -2.5\).

Step 1: Recognize Standard Form

The given inequality is in the standard form of a quadratic inequality: \(ax^2 + bx + c \leq 0\). Here, \(a = 4\), \(b = 20\), and \(c = 25\).

Step 2: Identify Roots of the Quadratic

To solve the inequality \(4x^2 + 20x + 25 = 0\), we will use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Calculate the discriminant \(b^2 - 4ac\): \(20^2 - 4 \times 4 \times 25 = 400 - 400 = 0\). This indicates a perfect square.

Step 3: Solve for the Root

Since the discriminant is zero, there is only one solution for the quadratic equation, namely \(x = \frac{-b}{2a}\). Substituting the values, we have \(x = \frac{-20}{2 \times 4} = \frac{-20}{8} = -2.5\).

Step 4: Analyze the Parabola

The quadratic expression \(4x^2 + 20x + 25\) is a perfect square trinomial: \((2x + 5)^2\). Since the only root is \(x = -2.5\) and \(a > 0\), the parabola opens upwards. Therefore, the inequality \((2x + 5)^2 \leq 0\) holds only for \(x = -2.5\).

Step 5: Conclude the Solution

The inequality \(4 x^2 + 20 x + 25 \leq 0\) is satisfied only for \(x = -2.5\). This is because at this point, the expression equals zero, and for any other point, it will be positive.

Key Concepts

Understanding the Standard Form of a Quadratic Equation

To begin solving quadratic inequalities like the one presented, it's crucial to identify that it's in the standard form of a quadratic equation. The standard form is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. In our specific problem \(4x^2 + 20x + 25 \leq 0\), we have the equation in standard form with coefficients identified as:

• \( a = 4 \)
• \( b = 20 \)
• \( c = 25 \)

Recognizing the standard form is the first step in solving quadratics as it allows us to utilize methods like factoring, completing the square, or employing the quadratic formula to find solutions. In inequalities, we analyze the zeros to understand the range of possible solutions.

Approaches to Solving Quadratic Equations

There are several ways to approach solving quadratic equations, each useful depending on the particular problem. Here's a brief overview:

• **Factoring:** Useful when the quadratic trinomial can be easily decomposed into the product of two binomials. It’s an efficient method when recognizable factor patterns exist.
• **Completing the Square:** This involves turning the quadratic equation into a perfect square trinomial, which can sometimes simplify the problem.
• **Quadratic Formula:** This is a universal method used when the equation cannot be easily factored. It’s applicable for any quadratic equation.

In our specific example, using the quadratic formula was necessary after observing that direct factoring was complex due to the lack of simple integer roots.

Using the Quadratic Formula

The quadratic formula is a reliable tool when factoring isn't straightforward. The formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). It allows us to find the roots of any quadratic equation by simply substituting the coefficients \(a\), \(b\), and \(c\) from the standard form.
In our situation with the equation \( 4x^2 + 20x + 25 = 0 \), we substitute:

• \( a = 4 \)
• \( b = 20 \)
• \( c = 25 \)

Computing the discriminant \( 20^2 - 4 \times 4 \times 25 = 400 - 400 = 0 \), we found it to be zero, indicating one unique solution: \( x = \frac{-20}{8} = -2.5 \). Thus, it provides a precise calculation of the root in cases where factoring is not apparent.

Exploring Discriminant Analysis

The discriminant provides valuable information about the number and type of roots in a quadratic equation. It is derived from the formula \( b^2 - 4ac \), where the term determines the nature of the roots:

• **If the discriminant is positive:** Two distinct real roots exist.
• **If it is zero:** There is exactly one real root, indicating the vertex of the parabola touches the x-axis.
• **If negative:** No real roots exist, suggesting the curve doesn't intersect the x-axis and solutions are complex numbers.

In the given problem, the discriminant \( 400 - 400 = 0 \), indicated a perfect square trinomial, leading us to only one solution \( x = -2.5 \). This root tells us that only this value satisfies \( 4x^2 + 20x + 25 \leq 0 \), proving crucial in the analysis of quadratic inequalities.