Problem 57 - College Algebra with Modeling and Visualization

Question

Use a table to solve. \(\frac{1}{8} x^{2}+x+2 \geq 0\)

Step-by-Step Solution

Answer

The inequality is satisfied for all real numbers \(x\).

Step 1: Identify the Quadratic Inequality

The given inequality is \( \frac{1}{8} x^2 + x + 2 \geq 0 \). This is a quadratic inequality because it includes an \(x^2\) term.

Step 2: Convert to Standard Form

Multiply the entire equation by 8 to eliminate the fraction. This yields \( x^2 + 8x + 16 \geq 0 \).

Step 3: Determine Roots of the Quadratic Equation

Consider the related quadratic equation \( x^2 + 8x + 16 = 0 \). Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \(a = 1\), \(b = 8\), and \(c = 16\).

Step 4: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\). For this equation, it is \(8^2 - 4(1)(16) = 64 - 64 = 0\). A discriminant of 0 means there is one real root.

Step 5: Find the Root

Since the discriminant is 0, the root of the equation is \( x = \frac{-8}{2(1)} = -4 \).

Step 6: Analyze the Inequality

The inequality \( x^2 + 8x + 16 \geq 0 \) signifies that the quadratic function is non-negative. Since the quadratic has a double root at \(x = -4\), the expression \((x + 4)^2 \geq 0\) is satisfied for all real numbers.

Step 7: Express the Solution

The inequality \( \frac{1}{8} x^2 + x + 2 \geq 0 \) holds for all \(x\) in the set of real numbers, as a perfect square \((x + 4)^2\) is always non-negative.

Key Concepts

Discriminant

The concept of the discriminant is crucial for understanding the nature of the roots of a quadratic equation. It is represented by the expression \(b^2 - 4ac\), derived from the coefficients \(a\), \(b\), and \(c\) in the standard quadratic equation \(ax^2 + bx + c = 0\). The discriminant provides information about the number and type of roots:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root, also known as a repeated or double root.
• A negative discriminant implies no real roots, but two complex roots.

In the exercise, the calculation showed a discriminant of 0, indicating a double root. This understanding helps assess the nature of the quadratic inequality solution.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of any quadratic equation in the form \(ax^2 + bx + c = 0\). This formula is expressed as \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. It's derived from completing the square on the general quadratic equation.

Let's break it down:

• "\(-b\)" means you're taking the opposite of \(b\).
• "\(\pm\sqrt{b^2 - 4ac}\)" indicates the two roots you'll get: one by adding, the other by subtracting the square root of the discriminant.
• "\(2a\)" means you divide the entire expression by \(2a\), which is doubled from \(a\).'"

In the context of our quadratic equation \(x^2 + 8x + 16 = 0\), substituting \(a = 1\), \(b = 8\), and \(c = 16\) gives \(x = \frac{-8 \pm \sqrt{0}}{2(1)} = -4\), confirming our sole real root.

Real Root

A real root refers to a solution of a quadratic equation that is a real number. Real roots can be visualized on a number line or graphically where the quadratic curve intersects the x-axis. Depending on the discriminant, you can have:

• No real roots (when the quadratic does not intersect the x-axis).
• One real root (tangent to the x-axis at one point, indicating the vertex of the parabola).
• Two real roots (curves that cross the x-axis twice).

For the exercise, the discriminant was 0, so the single real root was \(x = -4\). This means the quadratic expression \((x + 4)^2\) touches the x-axis exactly at \(x = -4\) without crossing it, indicating a vertex at this point.

Quadratic Function

Quadratic functions are polynomials of degree two, represented by \(f(x) = ax^2 + bx + c\). Their graph is a parabola, which can open upwards or downwards depending on the coefficient \(a\):

• If \(a > 0\), the parabola opens upwards, resembling a "U" shape.
• If \(a < 0\), it opens downwards, like an upside-down "U".

Important characteristics include the axis of symmetry, vertex, and direction of the opening.

Consider the quadratic function derived from the inequality \(x^2 + 8x + 16\). Here, \(a = 1\) indicates an upward-opening parabola with a perfect square form \((x + 4)^2\). Since this expression is always non-negative, solutions to the inequality \(x^2 + 8x + 16 \geq 0\) confirm the answer holds for all real numbers.