Problem 36 - Algebra 2

Question

Solve each inequality using a graph, a table, or algebraically. $$ 4 x^{2}+20 x+25 \geq 0 $$

Step-by-Step Solution

Answer

The solution is all real numbers, \( x \in \mathbb{R} \).

Step 1: Recognize the Inequality as a Quadratic

The given inequality \( 4x^2 + 20x + 25 \geq 0 \) is in quadratic form, \( ax^2 + bx + c \geq 0 \), where \( a = 4 \), \( b = 20 \), and \( c = 25 \). Our goal is to find the values of \( x \) for which this inequality holds true.

Step 2: Calculate the Discriminant

For a quadratic equation \( ax^2 + bx + c = 0 \), the discriminant is given by \( \Delta = b^2 - 4ac \). Substitute \( a = 4 \), \( b = 20 \), and \( c = 25 \) to find the discriminant:\[ \Delta = 20^2 - 4 \times 4 \times 25 = 400 - 400 = 0 \]Since the discriminant \( \Delta = 0 \), the quadratic has one real repeated root.

Step 3: Solve for the Root

Since \( \Delta = 0 \), use the quadratic formula \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \) to find the root:\[ x = \frac{-20 \pm \sqrt{0}}{2 \times 4} = \frac{-20}{8} = -2.5 \]Thus, \( x = -2.5 \) is the only root or zero of the quadratic \( 4x^2 + 20x + 25 = 0 \).

Step 4: Analyze the Parabola

Since the leading coefficient \( a = 4 \) is positive, the parabola opens upwards. If \( \Delta = 0 \), the quadratic expression \( 4x^2 + 20x + 25 \) reaches a minimum value of 0 at \( x = -2.5 \) and is non-negative for all \( x \).

Step 5: Determine the Solution Set

Because the expression is non-negative everywhere and equals zero at \( x = -2.5 \), the inequality holds true for all real \( x \):\[ x \in \mathbb{R} \]

Key Concepts

Discriminant

The discriminant is a key concept in understanding the nature of the roots of a quadratic equation. In the quadratic inequality \( 4x^2 + 20x + 25 \geq 0 \), we express the associated equation as \( ax^2 + bx + c = 0 \) to find its discriminant, \( \Delta \). This essential step helps us determine whether the quadratic has two distinct real roots, one real repeated root, or no real roots. The discriminant is calculated using the formula \( \Delta = b^2 - 4ac \). For this inequality, substituting \( a = 4 \), \( b = 20 \), and \( c = 25 \) gives us:

• \( b = 20 \)
• \( b^2 = 400 \)
• \( 4ac = 400 \)
• \( \Delta = 400 - 400 = 0 \)

When \( \Delta = 0 \), it indicates there is exactly one real repeated root. This tells us much about the nature of the quadratic's solution, especially in the context of an inequality. Understanding the role of the discriminant makes it easier to approach quadratic inequalities with confidence.

Real Roots

Real roots of a quadratic equation are the solutions for \( x \) where \( ax^2 + bx + c = 0 \). These roots can be found using the quadratic formula, and the discriminant informs us about their nature. In the quadratic \( 4x^2 + 20x + 25 \), because the discriminant is zero, we have one real repeated root. This means the parabola represented by the quadratic equation just touches the x-axis at a single point.It's important to remember:

• If \( \Delta > 0 \), there are two distinct real roots.
• If \( \Delta = 0 \), there is one real repeated root.
• If \( \Delta < 0 \), there are no real roots (the solutions are complex numbers).

For our equation, the real repeated root is calculated as \( x = -2.5 \). This tells us that the only place the parabola touches the x-axis is at \( x = -2.5 \), emphasizing the significance of the root in solving the inequality.

Quadratic Formula

The quadratic formula provides an efficient way to find the roots of any quadratic equation. Given by \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \), it's applicable to all quadratic equations in the standard form \( ax^2 + bx + c = 0 \). The "plus-minus" sign means there can be two solutions when \( \Delta > 0 \).In our particular equation \( 4x^2 + 20x + 25 \), because the discriminant is zero, the formula simplifies significantly:\[ x = \frac{-20 \pm \sqrt{0}}{2 \cdot 4} = \frac{-20}{8} = -2.5 \]This demonstrates how the quadratic formula works even when there's only one root. It effectively confirms that the quadratic touches the x-axis at this repeated root. The application of the quadratic formula here reinforces the understanding that it's a powerful tool for solving any quadratic, providing clear insights into the solution structure.

Parabola Analysis

Analyzing the parabola of a quadratic inequality gives us visual and analytical insights into the inequality's behavior. The equation \( 4x^2 + 20x + 25 \) represents a parabola. Given \( a = 4 \) is positive, this parabola opens upwards, confirming that any solutions to the inequality will vary depending on the position of the parabola's vertex in relation to the x-axis.Key features to note here include:

• Vertex: Since the discriminant \( \Delta = 0 \), it reaches its vertex at \( x = -2.5 \), which is also the minimum point.
• Minimum value: The quadratic reaches a minimum value of zero at the vertex, meaning at \( x = -2.5 \), the parabola just touches the x-axis.
• Behavior: As the parabola opens upwards and the expression is non-negative, the inequality \( 4x^2 + 20x + 25 \geq 0 \) holds true for all real numbers.

This parabola analysis shows visually and conceptually why every \( x \) satisfies the inequality, illustrating the completeness of the solution set \( x \in \mathbb{R} \). Such analyses aid greatly in understanding quadratic inequalities as a whole.