Solve each equation in Exercises 83–108 by the method of your choice. (2x + 3)(x + 4) = 1

Verified step by step guidance

1. Expand the left-hand side of the equation by using the distributive property: \((2x + 3)(x + 4)\). Multiply each term in \(2x + 3\) by each term in \(x + 4\). This will result in a quadratic expression.
2. Combine like terms from the expanded expression to simplify it into standard quadratic form: \(ax^2 + bx + c\).
3. Rewrite the equation so that all terms are on one side of the equation, setting it equal to zero. This will give you a standard quadratic equation: \(ax^2 + bx + c = 0\).
4. Choose a method to solve the quadratic equation. You can use factoring (if the quadratic is factorable), completing the square, or the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
5. Solve for \(x\) by applying the chosen method. If using the quadratic formula, substitute the values of \(a\), \(b\), and \(c\) into the formula and simplify to find the solutions for \(x\).

Key Concepts

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In the context of the given equation, recognizing that the left side is a product of two binomials allows for easier manipulation and solving of the equation.

Zero Product Property

The Zero Product Property states that if the product of two factors equals zero, then at least one of the factors must be zero. This principle is crucial when solving equations that have been factored, as it allows us to set each factor equal to zero to find the possible solutions for the variable.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax² + bx + c = 0, where a, b, and c are constants. The equation in the problem can be expanded and rearranged into a quadratic form, enabling the use of various methods such as factoring, completing the square, or the quadratic formula to find the solutions for x.