Problem 90 - Introductory Algebra for College Students

Question

If \((x+2)(x-4)=0\) indicates that \(x+2=0\) or \(x-4=0,\) explain why \((x+2)(x-4)=6\) does not mean \(x+2=6\) or \(x-4=6 .\) Could we solve the equation using \(x+2=3\) and \(x-4=2\) because \(3 \cdot 2=6 ?\)

Step-by-Step Solution

Answer

While the first equation \((x+2)(x-4)=0\) can be broken down into \(x+2=0\) or \(x-4=0,\) the same cannot be done for the second equation \((x+2)(x-4)=6\), as the product equals 6. Using \(x+2=3\) and \(x-4=2\) to solve, simply because \(3\cdot 2=6,\) is incorrect. Instead, the correct method of solving \((x+2)(x-4)=6\) would be to set the product of \(x+2\) and \(x-4\) equal to 6, and then solve for x.

Step 1: Understanding the first equation

The first equation, \((x+2)(x-4)=0,\) means the product of \((x+2)\) and \((x-4)\) is zero. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. So, this equation breaks down to \(x+2=0\) or \(x-4=0.\)

Step 2: Analyzing the incorrect breakdown of the second equation

The second equation is \((x+2)(x-4)=6.\) Unlike the first equation, the right side of this equation is not zero, but 6. Therefore, we cannot set \(x+2 = 6\) or \(x-4 = 6\) because the product of \(x+2\) and \(x-4\) equals 6, not \(x+2\) or \(x-4\). When a product of numbers equals a non-zero number, it does not imply that one or both of the numbers in the product equals that non-zero number.

Step 3: Responding to the proposed solution

Solving the equation using \(x+2=3\) and \(x-4=2\) because \(3\cdot 2=6 \) is incorrect. A product of two factors equals to their multiplier inner parts, not the factors themselves.

Step 4: Correct way to solve the second equation

In order to solve \((x+2)(x-4)=6,\) we need to set the product of \(x+2\) and \(x-4\) equal to 6, and then solve for x. Using FOIL method, we expand the left hand side to get \(x^2 -2x -8 = 6\), which upon simplifying becomes \(x^2 -2x -14=0 .\)

Step 5: Solving for x

Solving the quadratic equation \(x^2 -2x -14=0 \) we get two solutions for x using quadratic formula.