Solving 3x(x-1) - 4(x-5) = (x+3)^2 - 9 A Step-by-Step Guide

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This article provides a step-by-step guide on how to solve the quadratic equation 3x(x-1) - 4(x-5) = (x+3)^2 - 9. We will break down the process into manageable steps, ensuring a clear understanding of each operation. Our goal is to find the values of x that satisfy the equation, expressing the solutions as integers or simplified fractions. This comprehensive approach will not only solve the given equation but also provide a foundational understanding for tackling similar quadratic equations.

1. Expanding the Equation

To begin solving the equation, the first crucial step is to expand both sides of the equation. This involves distributing the terms and simplifying the expressions. Expanding the equation allows us to eliminate parentheses and combine like terms, bringing us closer to the standard form of a quadratic equation. Let's start with the left side of the equation: 3x(x-1) - 4(x-5). We distribute 3x across (x-1), which gives us 3x^2 - 3x. Next, we distribute -4 across (x-5), resulting in -4x + 20. Combining these, the left side becomes 3x^2 - 3x - 4x + 20.

Now, let’s move to the right side of the equation: (x+3)^2 - 9. To expand (x+3)^2, we can use the formula (a+b)^2 = a^2 + 2ab + b^2. Applying this, (x+3)^2 expands to x^2 + 6x + 9. Subtracting 9 from this gives us x^2 + 6x + 9 - 9, which simplifies to x^2 + 6x. So, the right side of the equation becomes x^2 + 6x. By expanding both sides of the equation, we have transformed the original equation into a more workable form, setting the stage for further simplification and solution.

2. Simplifying the Equation

Simplifying the equation is a vital step in solving for x. This process involves combining like terms on each side of the equation and rearranging the terms to bring the equation into a standard form. We'll start by simplifying the left side of the expanded equation, which is currently 3x^2 - 3x - 4x + 20. By combining the like terms -3x and -4x, we get -7x. Thus, the left side simplifies to 3x^2 - 7x + 20.

The right side of the equation is already in a simplified form: x^2 + 6x. Now, our equation looks like this: 3x^2 - 7x + 20 = x^2 + 6x. To further simplify and bring the equation to the standard quadratic form (ax^2 + bx + c = 0), we need to move all terms to one side of the equation. A common approach is to move all terms to the side with the larger x^2 coefficient, which in this case is the left side. We subtract x^2 from both sides, resulting in 2x^2 - 7x + 20 = 6x. Next, we subtract 6x from both sides to get 2x^2 - 7x - 6x + 20 = 0. Combining the -7x and -6x terms, we arrive at the simplified quadratic equation: 2x^2 - 13x + 20 = 0. This simplified form is crucial for applying various methods to solve for x, such as factoring, completing the square, or using the quadratic formula.

3. Factoring the Quadratic Equation

Factoring is a powerful technique for solving quadratic equations, provided the equation can be factored easily. This method involves expressing the quadratic expression as a product of two binomials. For our equation, 2x^2 - 13x + 20 = 0, we look for two binomials of the form (Ax + B)(Cx + D) such that their product equals the given quadratic expression. The key is to find two numbers that multiply to give the product of the leading coefficient (2) and the constant term (20), which is 40, and add up to the middle coefficient (-13).

After careful consideration, we find that the numbers -8 and -5 satisfy these conditions because (-8) * (-5) = 40 and (-8) + (-5) = -13. Now, we can rewrite the middle term of the quadratic equation using these numbers. The equation becomes 2x^2 - 8x - 5x + 20 = 0. Next, we factor by grouping. We group the first two terms and the last two terms: (2x^2 - 8x) + (-5x + 20) = 0. From the first group, we can factor out 2x, giving us 2x(x - 4). From the second group, we can factor out -5, giving us -5(x - 4). Now, the equation looks like this: 2x(x - 4) - 5(x - 4) = 0. Notice that (x - 4) is a common factor. We can factor it out, resulting in (2x - 5)(x - 4) = 0. By successfully factoring the quadratic equation, we have transformed it into a form where we can easily find the solutions for x.

4. Finding the Solutions for x

Once the quadratic equation is factored, finding the solutions for x becomes a straightforward process. The equation (2x - 5)(x - 4) = 0 tells us that the product of two factors 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. This principle allows us to set each factor equal to zero and solve for x individually.

First, we set the factor (2x - 5) equal to zero: 2x - 5 = 0. To solve for x, we add 5 to both sides, which gives us 2x = 5. Then, we divide both sides by 2, resulting in x = 5/2. This is one of the solutions for x. Next, we set the factor (x - 4) equal to zero: x - 4 = 0. To solve for x, we add 4 to both sides, which gives us x = 4. This is the second solution for x. Therefore, the solutions to the quadratic equation 2x^2 - 13x + 20 = 0 are x = 5/2 and x = 4. These values of x are the roots of the equation, meaning they are the points where the quadratic function intersects the x-axis.

5. Verification of Solutions

Verification is an essential step in the problem-solving process, especially when dealing with equations. It ensures that the solutions we've obtained are correct and satisfy the original equation. To verify our solutions, x = 5/2 and x = 4, we will substitute each value back into the original equation: 3x(x-1) - 4(x-5) = (x+3)^2 - 9. Let's start with x = 5/2.

Substituting x = 5/2 into the left side of the equation gives us: 3(5/2)((5/2)-1) - 4((5/2)-5). Simplifying inside the parentheses, we get 3(5/2)(3/2) - 4(-5/2). Further simplifying, this becomes (45/4) + (20/2), which equals (45/4) + (40/4) = 85/4. Now, let's substitute x = 5/2 into the right side of the equation: ((5/2)+3)^2 - 9. Simplifying inside the parentheses, we have (11/2)^2 - 9, which equals (121/4) - (36/4) = 85/4. Since both the left and right sides of the equation equal 85/4 when x = 5/2, this solution is verified.

Now, let's verify the solution x = 4. Substituting x = 4 into the left side of the equation gives us: 3(4)(4-1) - 4(4-5). Simplifying, we get 3(4)(3) - 4(-1), which equals 36 + 4 = 40. Substituting x = 4 into the right side of the equation gives us: (4+3)^2 - 9. Simplifying, we have (7)^2 - 9, which equals 49 - 9 = 40. Since both the left and right sides of the equation equal 40 when x = 4, this solution is also verified. By verifying both solutions, we can confidently state that x = 5/2 and x = 4 are the correct solutions to the equation 3x(x-1) - 4(x-5) = (x+3)^2 - 9.

6. Conclusion

In this article, we have successfully solved the quadratic equation 3x(x-1) - 4(x-5) = (x+3)^2 - 9 by following a step-by-step approach. First, we expanded both sides of the equation to eliminate parentheses and simplify the expressions. Next, we simplified the equation by combining like terms and rearranging them into the standard quadratic form. Then, we factored the quadratic equation, which allowed us to identify the solutions more easily. Following this, we found the solutions for x by applying the zero-product property. Finally, we verified our solutions by substituting them back into the original equation to ensure their accuracy. The solutions we found are x = 5/2 and x = 4. This comprehensive process not only provides the answers but also reinforces the fundamental techniques for solving quadratic equations, making it a valuable learning experience for anyone studying algebra. By understanding these steps, you can confidently tackle similar quadratic equations and deepen your mathematical skills.