Problem 38 - Algebra 2

Question

Solve each equation by factoring. \(x^{2}-3 x-10=0\)

Step-by-Step Solution

Answer

The solutions are \(x = 5\) and \(x = -2\).

Step 1: Identify the type of equation

The equation given is a quadratic equation of the form \(ax^2 + bx + c = 0\). In this case, \(a = 1\), \(b = -3\), and \(c = -10\).

Step 2: Factor the quadratic equation

To factor the quadratic equation \(x^2 - 3x - 10 = 0\), we need to find two numbers that multiply to \(-10\) (the constant term) and add to \(-3\) (the coefficient of \(x\)). These numbers are \(-5\) and \(2\). Thus, we can factor the equation as \((x - 5)(x + 2) = 0\).

Step 3: Solve for x using the Zero Product Property

Applying the Zero Product Property, which states if \((a)(b) = 0\), then \(a = 0\) or \(b = 0\), we set each factor equal to zero: \(x - 5 = 0\) or \(x + 2 = 0\).

Step 4: Find the solutions

Solve each equation from Step 3: \(x - 5 = 0\) gives \(x = 5\), and \(x + 2 = 0\) gives \(x = -2\). Thus, the solutions to the equation are \(x = 5\) and \(x = -2\).

Key Concepts

Quadratic Equations

Quadratic equations are fundamental expressions in algebra. They take the standard form of a polynomial equation expressed as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(x\) represents an unknown variable. In this form, \(a\) must not be zero, as this would eliminate the quadratic term \(x^2\). Quadratic equations graph as parabolas and can open upwards or downwards, depending on the sign of \(a\).
Understanding the structure is crucial:

• The "quadratic term" is represented by \(ax^2\).
• The "linear term" is \(bx\).
• The "constant term" is \(c\).

To solve these equations, you can use various methods such as factoring, completing the square, or using the quadratic formula. Each method is helpful in different scenarios, but factoring is often the most straightforward approach when possible.

Zero Product Property

The Zero Product Property is a powerful tool in algebra, particularly useful when working with quadratic equations. It states that if the product of two numbers (or expressions) equals zero, then at least one of the factors must be zero. Mathematically, if \((a)(b) = 0\), then \(a = 0\) or \(b = 0\).
In our exercise, this principle allows us to solve the equation once it is factored. After expressing the quadratic equation \(x^2 - 3x - 10 = 0\) as \((x - 5)(x + 2) = 0\), applying the Zero Product Property results in two simple linear equations:

• \(x - 5 = 0\)
• \(x + 2 = 0\)

Solving these linear equations provides the solutions to the original quadratic equation. This property streamlines the process, making it one of the most effective methods in solving quadratic equations.

Solving Equations Step-by-Step

Solving equations step-by-step ensures a clear understanding of the entire process. It involves breaking down the problem into manageable parts and tackling each systematically.
Here's how the solution unfolds for the quadratic equation \(x^2 - 3x - 10 = 0\):
1. **Identify the Type of Equation:** Recognize it as a quadratic equation. - Determine the values of \(a\), \(b\), and \(c\), which are 1, -3, and -10, respectively.
2. **Factor the Equation:** - Find two numbers that multiply to give the constant term \(-10\), and add up to the linear coefficient \(-3\). These numbers are \(-5\) and \(2\). - Write the equation as \((x - 5)(x + 2) = 0\).
3. **Apply the Zero Product Property:** - Set each factor equal to zero individually: \(x - 5 = 0\) and \(x + 2 = 0\).
4. **Solve for \(x\):** - From \(x - 5 = 0\), we find \(x = 5\). - From \(x + 2 = 0\), we find \(x = -2\).
The step-by-step approach helps ensure that every detail is understood, providing clarity and confidence in solving similar equations in the future.