Problem 43 - A Graphical Approach to College Algebra

Question

Solve each equation and inequality. (a) \(3\left(x^{2}+4\right)+2 x(3 x-12)=0\) (b) \(3\left(x^{2}+4\right)+2 x(3 x-12)<0\)

Step-by-Step Solution

Answer

(a) Solutions are \(x = 2\) and \(x = \frac{2}{3}\); (b) Solution is \(\frac{2}{3} < x < 2\).

Step 1: Expand and Simplify the Equation

The given equation is \(3(x^2 + 4) + 2x(3x - 12) = 0\). Expand each part: \(3x^2 + 12\) from the first term and \(6x^2 - 24x\) from the second term. Combine these to get \(3x^2 + 12 + 6x^2 - 24x = 0\). This simplifies to \(9x^2 - 24x + 12 = 0\).

Step 2: Solve the Quadratic Equation

The simplified equation is \(9x^2 - 24x + 12 = 0\). Use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 9\), \(b = -24\), and \(c = 12\). First, calculate the discriminant \(b^2 - 4ac = (-24)^2 - 4(9)(12) = 576 - 432 = 144\). Then, \(x = \frac{24 \pm \sqrt{144}}{18}\). The square root of 144 is 12, so \(x = \frac{24 \pm 12}{18}\). This results in two solutions: \(x = 2\) and \(x = \frac{2}{3}\).

Step 3: Solve the Inequality

The inequality is \(9x^2 - 24x + 12 < 0\). Use the solutions from the quadratic equation \(x = 2\) and \(x = \frac{2}{3}\) to determine intervals. The inequality \(9x^2 - 24x + 12\) is a quadratic expression which opens upwards (since \(a = 9 > 0\)). Thus, it's negative between the roots of the equation. Test intervals: \((-\infty, \frac{2}{3})\), \(\left(\frac{2}{3}, 2\right)\), and \((2, \infty)\). The inequality is satisfied in the interval \(rac{2}{3} < x < 2\).

Step 4: Finalize the Solutions

For the equation in part (a), the solutions are \(x = 2\) and \(x = \frac{2}{3}\). For the inequality in part (b), the solution is the range \(\frac{2}{3} < x < 2\).

Key Concepts

Quadratic Formula

The quadratic formula is one of the most powerful tools in solving quadratic equations. It's a standard approach used when equations cannot be easily factored or when other methods become cumbersome. This formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here, \(a\), \(b\), and \(c\) are coefficients from the quadratic equation's standard form: \(ax^2 + bx + c = 0\). The symbol \(\pm\) indicates that there will be two solutions, as most quadratic equations intersect the x-axis in two points.

• Make sure to correctly identify and substitute \(a\), \(b\), and \(c\) from your equation.
• Carefully perform calculations within the square root and division to avoid mistakes.

The quadratic formula ensures you can find solutions to any quadratic equation, given that you handle the discriminant part correctly.

Inequalities

Inequalities deal with expressions that are not strictly equal but instead greater than, less than, or belong to a range. In this context, they help us understand the range of values that satisfy an equation when expressions use symbols like \(<\) or \(>\). For example, the inequality \(9x^2 - 24x + 12 < 0\) enables us to find where the quadratic expression is negative.
When solving such inequalities, we:

• Identify the roots of the corresponding equation \(9x^2 - 24x + 12 = 0\) using methods like factorization or the quadratic formula.
• Test intervals defined by these roots to determine where the inequality holds.

This approach ensures that we evaluate each section between and outside of the identified roots, determining exact ranges where the quadratic is less than zero.

Discriminant

The discriminant is a component of the quadratic formula noted as \(b^2 - 4ac\). It plays a crucial role in determining the nature of the roots of a quadratic equation. Here's how you can interpret the discriminant:

• If the discriminant \(b^2 - 4ac > 0\), the equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), there is one real root, often referred to as a repeated or double root.
• If \(b^2 - 4ac < 0\), the roots are complex and not real.

For the equation \(9x^2 - 24x + 12 = 0\), we calculated the discriminant as \(144\), indicating two distinct real solutions. Understanding the discriminant allows you to predict the type of solutions before actually finding them.

Factorization

Factorization is a method often used to solve quadratic equations, depending on their simplicity or structure. The goal is to express the quadratic as a product of two linear equations. For example, if an equation can be expressed like \((x - p)(x - q) = 0\), then \(x = p\) and \(x = q\) are the solutions.
Steps to factor a quadratic:

• Attempt to rewrite the quadratic in a token format, identifying patterns or easy simplifications.
• Check if it's possible to easily express it into the form \((mx + n)(px + q) = 0\).
• If factorization is not straightforward, use the quadratic formula to ensure solutions.

In our original equation, factorization might not be the easiest route, which is why using the quadratic formula was more straightforward. Factorization becomes handier when dealing with simpler quadratic structures or recognizable patterns.