Use the method described in Exercises 83–86, if applicable, and properties of absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can be solved by inspection.) | 3x² - 14x | = 5

Verified step by step guidance

Recognize that the equation involves an absolute value: $|3x^2 - 14x| = 5$. The definition of absolute value tells us that $|A| = B$ means $A = B$ or $A = -B$ when $B \geq 0$.

Set up two separate equations based on the absolute value property: $3x^2 - 14x = 5$ and $3x^2 - 14x = -5$.

Solve each quadratic equation separately. For the first equation $3x^2 - 14x - 5 = 0$, use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a=3$, $b=-14$, and $c=-5$.

Similarly, solve the second quadratic equation $3x^2 - 14x + 5 = 0$ using the quadratic formula with $a=3$, $b=-14$, and $c=5$.

After finding the solutions from both quadratics, check each solution by substituting back into the original absolute value equation to ensure they satisfy $|3x^2 - 14x| = 5$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Absolute Value Equations

An absolute value equation involves expressions within absolute value bars, which represent the distance from zero on the number line. To solve |A| = B, where B ≥ 0, we set A = B and A = -B, creating two separate equations to solve. Understanding this principle is essential for breaking down the given equation.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically in the form ax² + bx + c = 0. Solving them involves factoring, completing the square, or using the quadratic formula. Since the expression inside the absolute value is quadratic, solving the resulting equations requires knowledge of these methods.

Properties of Absolute Value Inequalities

Properties of absolute value inequalities help in solving equations or inequalities involving absolute values by considering cases based on the definition of absolute value. For example, |X| = k implies X = k or X = -k, and understanding these properties aids in correctly setting up and solving the problem.

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Textbook Question

Use the method described in Exercises 83–86, if applicable, and properties of absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can be solved by inspection.) | 3x² + x | = 14

Textbook Question

Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (Do not solve the equation.) See Example 9.

$9 x^{2} + 11 x + 4 = 0$