Solve each equation in Exercises 83–108 by the method of your choice. (2x - 5)(x + 1) = 2

Verified step by step guidance

1. Rewrite the equation in standard form by expanding the left-hand side and moving all terms to one side of the equation. Start by distributing the terms: \((2x - 5)(x + 1) = 2\). Expand \((2x - 5)(x + 1)\) to get \(2x^2 + 2x - 5x - 5 = 2\). Combine like terms to simplify: \(2x^2 - 3x - 5 = 2\). Subtract 2 from both sides to set the equation to zero: \(2x^2 - 3x - 7 = 0\).
2. Identify the type of equation. This is a quadratic equation in the form \(ax^2 + bx + c = 0\), where \(a = 2\), \(b = -3\), and \(c = -7\).
3. Choose a method to solve the quadratic equation. You can use factoring (if possible), completing the square, or the quadratic formula. Since factoring may not be straightforward here, the quadratic formula is a good choice: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
4. Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula. Replace \(a = 2\), \(b = -3\), and \(c = -7\) into \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This becomes \(x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-7)}}{2(2)}\).
5. Simplify the expression under the square root (the discriminant) and the rest of the formula. Compute \((-3)^2 - 4(2)(-7)\) and simplify the denominator \(2(2)\). Then, simplify the entire expression to find the two possible 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.

Isolating the Variable

Isolating the variable involves rearranging an equation to get the variable on one side and the constants on the other. This technique is essential for solving equations, as it simplifies the process of finding the value of the variable, especially after applying methods like factoring or using the quadratic formula.