Solving (x+1)(x-3)=12 Which Statement Is True
In this comprehensive guide, we will delve into the problem of solving the equation (x+1)(x-3)=12. This type of equation, involving the product of binomials, frequently appears in algebra and requires a systematic approach to find its solutions. We will explore the steps involved in simplifying the equation, identifying the correct method for solving it, and interpreting the results. Understanding how to solve such equations is fundamental for success in various mathematical disciplines.
Understanding the Problem
To begin, let’s clearly understand the problem we are tackling. The equation given is (x+1)(x-3)=12. This is a quadratic equation disguised in factored form. The goal is to find the values of x that satisfy this equation. A common mistake is to directly set each factor equal to 12, which is what option A suggests. However, this is incorrect because the equation states that the product of the factors equals 12, not that each factor individually equals 12. Similarly, option B suggests setting each factor equal to zero, which would be valid if the right-hand side of the equation were zero, but it is not.
The Pitfalls of Incorrect Approaches
It's crucial to avoid the common traps that students often fall into when faced with this type of equation. Directly equating each factor to 12 or 0 stems from a misunderstanding of the zero-product property, which only applies when the product of factors equals zero. For instance, if we had the equation (x+1)(x-3)=0, then we could correctly deduce that either x+1=0 or x-3=0. But since our equation equals 12, we must first manipulate the equation to fit the standard quadratic form before we can solve it effectively. This involves expanding the product, simplifying the expression, and rearranging the terms to set the equation equal to zero. This careful algebraic manipulation is the cornerstone of solving quadratic equations accurately.
Steps to Solve the Equation (x+1)(x-3)=12
To correctly solve the equation (x+1)(x-3)=12, we need to follow a series of steps that involve expanding the product, simplifying the equation, and rearranging the terms. This process will transform the equation into a standard quadratic form, which we can then solve using factoring or the quadratic formula.
1. Expand the Product
The first step is to expand the left side of the equation using the distributive property (also known as the FOIL method). This involves multiplying each term in the first binomial by each term in the second binomial:
(x+1)(x-3) = x(x) + x(-3) + 1(x) + 1(-3)
Simplifying this gives us:
x^2 - 3x + x - 3
Combining like terms, we get:
x^2 - 2x - 3
2. Rewrite the Equation in Standard Quadratic Form
Now we substitute the expanded form back into the original equation:
x^2 - 2x - 3 = 12
To put the equation in standard quadratic form (ax^2 + bx + c = 0), we need to subtract 12 from both sides:
x^2 - 2x - 3 - 12 = 0
This simplifies to:
x^2 - 2x - 15 = 0
3. Factor the Quadratic Equation
The next step is to factor the quadratic expression. We are looking for two numbers that multiply to -15 and add to -2. These numbers are -5 and 3. So, we can factor the quadratic as follows:
(x - 5)(x + 3) = 0
4. Apply the Zero-Product Property
Now we can apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This gives us two separate equations to solve:
x - 5 = 0 or x + 3 = 0
5. Solve for x
Solving each equation for x, we get:
For x - 5 = 0:
x = 5
For x + 3 = 0:
x = -3
Thus, the solutions to the equation are x = 5 and x = -3.
Analyzing the Answer Choices
Now that we have found the solutions to the equation, we can evaluate the given answer choices to determine which statement is true.
Option A: x+1=12 or x-3=12
This option is incorrect. As we discussed earlier, it is a common mistake to directly equate each factor to 12. This approach does not follow the correct algebraic procedure for solving quadratic equations.
Option B: x+1=0 or x-3=0
This option is also incorrect. This approach would be valid if the original equation were (x+1)(x-3)=0, but it is not. The equation is equal to 12, so we must first rewrite the equation in standard quadratic form before solving.
Option C: x-5=0 or x+3=0
This option is correct. These equations correspond to the factors we found when we factored the quadratic equation: (x - 5)(x + 3) = 0. Setting each factor equal to zero gives us the correct solutions for x.
Conclusion
In conclusion, the correct answer is C. x-5=0 or x+3=0. We arrived at this answer by correctly expanding and simplifying the original equation, rewriting it in standard quadratic form, factoring the quadratic, and applying the zero-product property. This systematic approach is essential for solving quadratic equations and avoiding common mistakes. Understanding these steps will not only help in solving this particular problem but will also provide a strong foundation for tackling more complex algebraic problems in the future.
By following the steps outlined in this guide, you can confidently solve quadratic equations of this type. Remember to expand and simplify, rewrite in standard form, factor (or use the quadratic formula if factoring is not possible), and apply the zero-product property. With practice, these techniques will become second nature, enabling you to excel in your mathematical studies.
