Finding Real Solutions Product For (x^2 - 5x + 7)^2 - (x - 3) * (x - 2) - 3 = 0
Find the product of all real solutions to the equation (x^2 - 5x + 7)^2 - (x - 3) * (x - 2) - 3 = 0.
In this article, we will delve into solving a fascinating algebraic equation and finding the product of its real solutions. The equation at hand is: (x^2 - 5x + 7)^2 - (x - 3) * (x - 2) - 3 = 0. This problem challenges our algebraic skills, requiring us to simplify, manipulate, and ultimately solve for the values of x that satisfy the equation. Let's embark on this mathematical journey together!
Understanding the Problem
The first step in tackling any complex equation is to thoroughly understand the problem. Our main goal is to find all the real numbers 'x' that make the equation true. Once we have those numbers, we need to multiply them together. This final product is what we are looking for. The given equation combines polynomial expressions and requires careful simplification to solve it efficiently. We need to expand the terms, combine like terms, and hopefully arrive at a more manageable form, possibly a quadratic equation, which we can then solve using standard methods.
Solving the Equation
To solve the equation (x^2 - 5x + 7)^2 - (x - 3) * (x - 2) - 3 = 0, we will follow a step-by-step process:
Step 1: Expand the terms
Let's start by expanding each part of the equation. First, we expand (x^2 - 5x + 7)^2. This means multiplying (x^2 - 5x + 7) by itself:
(x^2 - 5x + 7)^2 = (x^2 - 5x + 7) * (x^2 - 5x + 7)
Expanding this, we get:
x^4 - 5x^3 + 7x^2 - 5x^3 + 25x^2 - 35x + 7x^2 - 35x + 49
Combining like terms, we have:
x^4 - 10x^3 + 39x^2 - 70x + 49
Next, we expand (x - 3) * (x - 2):
(x - 3) * (x - 2) = x^2 - 2x - 3x + 6
Simplifying, we get:
x^2 - 5x + 6
Step 2: Substitute the expanded terms into the equation
Now, let's substitute these expanded terms back into the original equation:
(x^4 - 10x^3 + 39x^2 - 70x + 49) - (x^2 - 5x + 6) - 3 = 0
Step 3: Simplify the equation
Remove the parentheses and combine like terms:
x^4 - 10x^3 + 39x^2 - 70x + 49 - x^2 + 5x - 6 - 3 = 0
Combining like terms, we get:
x^4 - 10x^3 + 38x^2 - 65x + 40 = 0
Step 4: Look for a possible substitution
This quartic equation looks complex, but we might be able to simplify it further. Let's go back to the original equation and see if we can make a useful substitution.
(x^2 - 5x + 7)^2 - (x - 3) * (x - 2) - 3 = 0
Notice that (x - 3) * (x - 2) = x^2 - 5x + 6. Let's rewrite the original equation using this observation:
(x^2 - 5x + 7)^2 - (x^2 - 5x + 6) - 3 = 0
Now, let's make a substitution. Let y = x^2 - 5x. Then, the equation becomes:
(y + 7)^2 - (y + 6) - 3 = 0
Step 5: Expand and simplify the new equation
Expanding (y + 7)^2, we get:
y^2 + 14y + 49
Substituting back into the equation:
y^2 + 14y + 49 - (y + 6) - 3 = 0
Simplifying, we get:
y^2 + 14y + 49 - y - 6 - 3 = 0
y^2 + 13y + 40 = 0
Step 6: Solve the quadratic equation for y
Now we have a quadratic equation in terms of y. We can solve it by factoring:
y^2 + 13y + 40 = 0
(y + 5)(y + 8) = 0
So, the solutions for y are:
y = -5 or y = -8
Step 7: Substitute back to find x
Now we need to substitute back to find the values of x. Remember that we let y = x^2 - 5x.
Case 1: y = -5
x^2 - 5x = -5
x^2 - 5x + 5 = 0
We can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -5, and c = 5.
x = (5 ± √((-5)^2 - 4 * 1 * 5)) / 2 * 1
x = (5 ± √(25 - 20)) / 2
x = (5 ± √5) / 2
So, the two solutions for x in this case are:
x_1 = (5 + √5) / 2
x_2 = (5 - √5) / 2
Case 2: y = -8
x^2 - 5x = -8
x^2 - 5x + 8 = 0
Using the quadratic formula again, with a = 1, b = -5, and c = 8:
x = (5 ± √((-5)^2 - 4 * 1 * 8)) / 2 * 1
x = (5 ± √(25 - 32)) / 2
x = (5 ± √(-7)) / 2
Since we have a negative number under the square root, these solutions are complex (not real). Therefore, we don't include them in our final product.
Step 8: Find the product of the real solutions
We found two real solutions for x:
x_1 = (5 + √5) / 2
x_2 = (5 - √5) / 2
To find the product, we multiply these two solutions:
Product = x_1 * x_2 = ((5 + √5) / 2) * ((5 - √5) / 2)
Using the difference of squares formula (a + b)(a - b) = a^2 - b^2:
Product = (5^2 - (√5)^2) / 4
Product = (25 - 5) / 4
Product = 20 / 4
Product = 5
Final Answer
The product of all real solutions to the equation (x^2 - 5x + 7)^2 - (x - 3) * (x - 2) - 3 = 0 is 5. So the correct answer is C) 5.
This detailed solution demonstrates the power of algebraic manipulation and the importance of methodical problem-solving. By breaking down a complex problem into smaller, manageable steps, we can arrive at the correct answer with confidence.
Summary of Steps
To recap, here are the key steps we took to solve this problem:
- Expand the terms: We expanded both (x^2 - 5x + 7)^2 and (x - 3) * (x - 2).
- Substitute the expanded terms: We substituted the expanded terms back into the original equation.
- Simplify the equation: We combined like terms to simplify the equation.
- Look for a possible substitution: We recognized a pattern that allowed us to substitute y = x^2 - 5x.
- Expand and simplify the new equation: We expanded and simplified the resulting equation in terms of y.
- Solve the quadratic equation for y: We solved the quadratic equation to find the values of y.
- Substitute back to find x: We substituted the values of y back into the equation y = x^2 - 5x to find the values of x.
- Find the product of the real solutions: We identified the real solutions and multiplied them together to find the final product.
By following these steps, we were able to successfully solve the problem and find the product of the real solutions.
Importance of Algebraic Skills
This problem highlights the importance of strong algebraic skills. The ability to expand expressions, simplify equations, and make strategic substitutions is crucial for solving complex mathematical problems. Practice and familiarity with these techniques can greatly improve your problem-solving abilities.
Algebraic equations like this are not just abstract exercises; they represent real-world relationships and can be used to model and solve problems in various fields, including physics, engineering, and economics. Therefore, mastering these skills is an investment in your future success.
Further Practice
To further hone your algebraic skills, consider tackling similar problems. Look for equations that involve polynomial expressions, substitutions, and quadratic formulas. The more you practice, the more confident and proficient you will become.
Here are some suggestions for further practice:
- Solve other quartic equations.
- Practice making substitutions to simplify equations.
- Work through various quadratic equation problems.
- Explore problems that require factoring and expanding expressions.
By dedicating time to practice and develop your algebraic skills, you will not only improve your ability to solve equations but also enhance your overall mathematical thinking and problem-solving abilities.
Conclusion
In conclusion, finding the product of real solutions to the equation (x^2 - 5x + 7)^2 - (x - 3) * (x - 2) - 3 = 0 required a careful step-by-step approach, strategic substitution, and a solid understanding of algebraic principles. The final answer, 5, demonstrates the power of methodical problem-solving and the importance of algebraic skills. Keep practicing, keep exploring, and keep challenging yourself with mathematical problems!