Problem 58 - Elementary Algebra

Question

Solve. $$ (x-1)(x-10)=22 $$

Step-by-Step Solution

Answer

x = 12 or x = -1.

Step 1: Expand the Equation

First, we need to expand the left side of the equation using the distributive property: \((x-1)(x-10) = x(x-10) - 1(x-10)\). This simplifies to \(x^2 - 10x - x + 10 = x^2 - 11x + 10\). The equation is now \(x^2 - 11x + 10 = 22\).

Step 2: Set the Equation to Zero

Subtract 22 from both sides to set the equation to zero:\(x^2 - 11x + 10 - 22 = 0\), which simplifies to \(x^2 - 11x - 12 = 0\).

Step 3: Factor the Quadratic Equation

To factor \(x^2 - 11x - 12 = 0\), look for two numbers that multiply to \(-12\) and add to \(-11\). These numbers are \(-12\) and \(1\): \((x - 12)(x + 1) = 0\).

Step 4: Solve for x

Set each factor equal to zero and solve for \(x\): 1. \(x - 12 = 0\) gives \(x = 12\).2. \(x + 1 = 0\) gives \(x = -1\).So, the solutions are \(x = 12\) and \(x = -1\).

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions. It allows you to multiply a single term by each term inside a parenthesis. In the equation

• \((x-1)(x-10)\), the distributive property helps by breaking it into two parts: \(x(x-10)\) and \(-1(x-10)\).
• This results in \(x^2 - 10x - x + 10\).

Now the expression inside the parentheses is expanded. You can always double-check the expanded form by multiplying back to ensure no mistakes were made. Thus, you end up with a simpler equation to work with: \(x^2 - 11x + 10 = 22\). Remember, accurately applying the distributive property is crucial for solving equations effectively.

Factoring Quadratic Equations

Factoring quadratic equations is a technique used to express a quadratic expression as the product of two binomials. It is an essential skill when solving quadratic equations. Consider the equation we have after using the distributive property:

• \(x^2 - 11x - 12 = 0\).

To factor this, identify two numbers that multiply to give the constant term,

• \(-12\), and add up to the linear coefficient \(-11\).

These numbers are \(-12\) and \(1\). Substituting these back in, we have the factors

• \((x - 12)(x + 1)\).

If you're unsure about the numbers, list the factors of the constant term until you find a pair that satisfies both conditions. Each time you factor, you are simplifying the process of finding solutions.

Finding Solutions of Quadratic Equations

Once the quadratic expression is factored, finding the solutions of the equation becomes straightforward. You'll apply the Zero Product Property, which states that if the product of two numbers is zero, then at least one of the numbers must be zero. Using this property:

• Set each factor to zero: \(x - 12 = 0\) and \(x + 1 = 0\).
• Solving these gives you \(x = 12\) and \(x = -1\).

Therefore, the solutions to the equation \((x-1)(x-10) = 22\) are \(x = 12\) and \(x = -1\). Verify your solutions by substituting them back into the original equation if needed. This ensures your solution is correct and gives you a clear understanding of solving quadratic equations efficiently.