Problem 125 - College Algebra

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

No, we cannot say x+2=6 or x-4=6 despite (x+2)(x-4)=6. This reasoning misunderstands the zero-product property, where the decomposition used relies on the product being zero. For (x+2)(x-4)=6 you need to apply standard mathematical method such as quadratic formula to solve it, rather than decomposing into two separate equations. The solution x+2=3 and x-4=2 because 3 * 2 = 6 doesn't hold because these equations aren't the actual factorizations of the original equation anyways.

Step 1: Understand the Equations

In the equation (x+2)(x-4)=0, this can be decomposed into x+2 = 0 or x-4 = 0 because of the zero product property, which states that if a product of factors is zero, then at least one of the factors must be zero. Therefore, the solution is x = -2 or x = 4.

Step 2: Responding to the Query

For the equation (x+2)(x-4)=6, this can't be simplified to x+2=6 or x-4=6. Because if we substitute x=4 from x+2=6, we will get 6 = 6, but substituting x=4 into x-4=6 we will get 0=6, which contradicts the initial equation.

Step 3: Why The Technique Fails

Therefore, this approach can't be used to solve the equation (x+2)(x-4)=6. The zero-product property can't be applied here because the product isn't zero.

Step 4: Correct Approach to Solve (x+2)(x-4)=6

To correctly solve the equation (x+2)(x-4)=6, we need to expand the brackets to obtain \(x^2 -2x - 8 = 6\). Solving the equation \(x^2 -2x - 14 = 0\), we can apply the quadratic formula \(x = \frac{-b± \sqrt{b^2 - 4ac}}{2a}\) to find the solutions for x.

Step 5: Exploring Suggested Solution

As for the suggestion to use x+2=3 and x-4=2 because 3 \cdot 2 = 6, this doesn't hold because these equations were derived arbitrarily and aren't the actual factorizations of the original expression (x+2)(x-4).

Key Concepts

Solving Quadratic Equations

Quadratic equations are a staple in algebra and can be recognized by their standard form, which looks like this: \(ax^2 + bx + c = 0\). To solve a quadratic equation, you have a few options available. One of them is factoring the quadratic into a product of binomials if possible, as with \(x^2 -2x - 14 = 0\). Sometimes, however, the equation does not neatly factor, or it may be time-consuming to find the right factors.
In these cases, we can use the quadratic formula, which is a surefire way to find the roots of any quadratic equation when a, b, and c are known. But before you jump to the quadratic formula, it's important to assess if simpler methods, like factoring or completing the square, are applicable. Solving quadratic equations is a fundamental skill that helps in understanding more complex algebraic equations and functions.

Quadratic Formula

The Quadratic Formula is a mathematical lifeline when other methods of solving quadratics, such as factoring, are not suitable. This formula is represented as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
Let’s break it down. To use this tool, you'll need to identify the coefficients a, b, and c from the standard form of the quadratic equation. Then, insert these values into the formula, making sure to perform each mathematical operation with care. The \(\pm\) sign indicates that there will be two solutions, also known as roots of the equation — one where you add the square root, and one where you subtract it. The beauty of the quadratic formula lies in its reliability; it works every time, regardless of whether the quadratic is factorable or not.

Factorization

Factorization involves breaking down a complex expression into a product of simpler ones. In algebra, it is often used with polynomials such as quadratic equations. For instance, the quadratic equation \(x^2 -2x - 14 = 0\) can be factored into \(x+2)(x-7)=0\), if we find the two numbers that multiply to give -14 and add up to -2.
Once you have the factors, the next step is exploiting the zero product property. According to this property, if the product of two factors equals zero, then at least one of the factors must be zero. This allows us to set each factor equal to zero and solve for the variable. Factorization is a powerful tool, but it works best when the roots are rational numbers. If the roots are irrational or complex, the quadratic formula might be a more straightforward option.

Algebraic Properties

Algebraic properties are the rules that govern algebraic operations and expressions. They are the backbone of solving equations and manipulating expressions. Key properties include the Commutative Property (you can change the order of numbers in addition or multiplication), the Associative Property (regroups numbers and it won't change the result), the Distributive Property (lets you multiply a number by a sum or difference), and the Additive Inverse Property (every number has an opposite, which they sum to 0).
In the context of our example, these properties help us expand brackets and combine like terms in the process of solving quadratic equations. Understanding and correctly applying these properties can simplify complex algebraic processes and lead to correct solutions.