Problem 43 - A Graphical Approach to Precalculus with Limits

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) The solutions are \(x = 2\) or \(\frac{2}{3}\). (b) The inequality holds for \(\frac{2}{3} < x < 2\).

Step 1: Expand and Simplify the Equation

Begin by expanding the expression by distributing the terms: 1. Expand both parts: - The first term: \[ 3(x^2 + 4) = 3x^2 + 12 \] - The second term: \[ 2x(3x - 12) = 6x^2 - 24x \] 2. Combine these results to form the equation: \[ 3x^2 + 12 + 6x^2 - 24x = 0 \]

Step 2: Combine Like Terms

Combine like terms in the equation: 1. Combine the \(x^2\) terms: \[ 3x^2 + 6x^2 = 9x^2 \] 2. Write the new equation: \[ 9x^2 - 24x + 12 = 0 \]

Step 3: Solve the Quadratic Equation (Part a)

Use the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), to solve for \(x\) in \(9x^2 - 24x + 12 = 0\):1. Identify coefficients: \(a = 9\), \(b = -24\), \(c = 12\). 2. Calculate the discriminant: \[ b^2 - 4ac = (-24)^2 - 4 \cdot 9 \cdot 12 = 576 - 432 = 144 \]3. Calculate \(x\): \[ x = \frac{-(-24) \pm \sqrt{144}}{2 \cdot 9} = \frac{24 \pm 12}{18} \] - For \(24 + 12\): \[ x = \frac{36}{18} = 2 \] - For \(24 - 12\): \[ x = \frac{12}{18} = \frac{2}{3} \] So, \(x = 2\) or \(x = \frac{2}{3}\).

Step 4: Translate to Inequality (Part b) and Solve

Now, solve the inequality \(9x^2 - 24x + 12 < 0\): 1. Determine when the quadratic is less than zero by finding intervals between and outside the roots found: - The roots from Part a are \(x = \frac{2}{3}\) and \(x = 2\).2. Test intervals around the roots: - Pick a value less than \(\frac{2}{3}\), like \(x = 0\) in \(9x^2 - 24x + 12\) gives a positive result (12). - Pick a value between \(\frac{2}{3}\) and 2, like \(x = 1\). Plugging in gives \(-3\), which is negative. - Pick a value greater than 2, like \(x = 3\). Plugging in gives \(21\), which is positive.3. The inequality holds in the interval: \(\frac{2}{3} < x < 2\).

Step 5: Solution Summary

For part (a), the solutions to the equation are \(x = 2\) or \(x = \frac{2}{3}\). For part (b), the inequality is true for values of \(x\) within the interval \(\frac{2}{3} < x < 2\).

Key Concepts

Inequalities

An inequality is a mathematical expression that shows the relation between two expressions that are not equal. In our given problem, after solving the quadratic equation, we also needed to solve an inequality: \(9x^2 - 24x + 12 < 0\).

Unlike equations, which find specific values of \(x\), inequalities identify the range or set of values that make the inequality true. This involves checking the intervals between and around the roots of the corresponding quadratic equation.

• Testing Values: Choose test points from different intervals based on the roots.
• Sign Change: If substitution results in a negative, it falls within the inequality range.

Understanding inequalities is crucial because they explain not just the solution, but the entire range of solutions, providing a more comprehensive view of possible scenarios.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of quadratic equations, often in the form \(ax^2 + bx + c = 0\). In our problem, we had the equation \(9x^2 - 24x + 12 = 0\) which we solved using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 9\), \(b = -24\), and \(c = 12\). The term \(b^2 - 4ac\) is known as the discriminant and tells us about the nature of the roots:

• If positive, the equation has two real and distinct roots.
• If zero, the equation has exactly one real root.
• If negative, there are no real roots, only complex ones.

In our case, the discriminant was \(144\), indicating two real roots at \(x = 2\) and \(x = \frac{2}{3}\). Thus, the quadratic formula not only gives precise solutions but also indicates how many solutions exist and their nature.

Polynomial Simplification

Polynomial simplification involves making a polynomial less complex while keeping the same identity/value. In our problem, we started with a complex expression \(3(x^2 + 4) + 2x(3x - 12)\), which required simplification.

Simplifying polynomials involves two key steps:

• Expanding: We distribute each term across the expression.
• Combining like terms: We combine terms that have the same variable part.

By performing these steps, we transformed the original expression into a clearer form \(9x^2 - 24x + 12 = 0\). Simplifying polynomials is a necessary step in solving many algebraic problems, as it puts equations or expressions into a manageable form.

Roots and Intervals

Roots of a quadratic equation are the values of \(x\) that make the equation zero. Solving our quadratic equation \(9x^2 - 24x + 12 = 0\), we found the roots \(x = 2\) and \(x = \frac{2}{3}\). These roots are also key to solving related inequalities.

To solve the inequality \(9x^2 - 24x + 12 < 0\), roots help us determine intervals to test:

• Points Before, Between, and After Roots: Test each interval with a point to see if it satisfies the inequality.
• Understanding Sign Changes: Intervals where the expression changes sign (i.e., positive to negative) indicate parts of the solution set.

Through testing, we found the interval \(\frac{2}{3} < x < 2\), where the inequality holds true. This process of finding roots and evaluating intervals allows us to not only solve equations precisely but also understand how they behave across different values, essential for mastering algebraic inequalities.