Solving X² - 7x + 10 = 0 Find The Solution Set
Hey guys! Let's dive into solving this quadratic equation together. We've got x² - 7x + 10 = 0, and we need to find the solution set. Don't worry, it's not as scary as it looks! We'll break it down step by step.
Understanding Quadratic Equations
Before we jump into solving, let's quickly recap what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. That means the highest power of the variable (in this case, 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and 'a' is not equal to zero. If 'a' were zero, the term with x² would disappear, and we'd be left with a linear equation instead.
In our equation, x² - 7x + 10 = 0, we can identify the coefficients: a = 1, b = -7, and c = 10. Recognizing these coefficients is the first step towards solving the equation. Now, there are several methods we can use to find the solutions (also called roots) of a quadratic equation. The most common methods are factoring, using the quadratic formula, and completing the square. For this particular equation, factoring is the easiest and fastest way to go, so let's use that.
Factoring involves rewriting the quadratic expression as a product of two binomials. Think of it like reversing the process of expanding two brackets. We need to find two numbers that, when multiplied, give us 'c' (which is 10), and when added, give us 'b' (which is -7). This might sound tricky, but with a bit of practice, it becomes second nature. Once we've factored the equation, we can use the zero-product property to find the solutions. This property states that if the product of two factors is zero, then at least one of the factors must be zero. This is the key to unlocking the solutions to our equation.
The Factoring Method
The factoring method is a technique used to solve quadratic equations by expressing the quadratic expression as a product of two linear factors. This method is particularly efficient when the quadratic equation can be easily factored, which is the case for our equation, x² - 7x + 10 = 0. The underlying principle of factoring relies on reversing the distributive property (or the FOIL method) that we use to expand products of binomials.
In our specific example, we need to find two numbers that multiply to give us the constant term, which is 10, and add up to give us the coefficient of the linear term, which is -7. Let's think about the factors of 10. We have 1 and 10, and 2 and 5. Since we need the sum to be -7, we'll need to consider negative factors. The pair -2 and -5 fits the bill perfectly because -2 * -5 = 10 and -2 + -5 = -7. So, we can rewrite the middle term (-7x) as -2x - 5x, which allows us to factor by grouping.
Now, we rewrite the equation as x² - 2x - 5x + 10 = 0. Next, we group the terms in pairs: (x² - 2x) + (-5x + 10) = 0. From the first group, we can factor out an 'x', which gives us x(x - 2). From the second group, we can factor out a -5, which gives us -5(x - 2). Notice that we now have a common factor of (x - 2) in both terms. This is a crucial step because it allows us to factor the expression further.
We can factor out the (x - 2) from the entire equation, resulting in (x - 2)(x - 5) = 0. Now we've successfully factored the quadratic equation into two linear factors. The next step is to apply the zero-product property, which is the cornerstone of solving factored equations. This property allows us to transform a single equation into two simpler equations, each of which is easy to solve. By understanding and applying the factoring method, we've made significant progress towards finding the solutions to our quadratic equation.
Applying the Zero-Product Property
The zero-product property is a fundamental principle in algebra that states: if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. In simpler terms, if we have A * B = 0, then either A = 0, B = 0, or both A and B are zero. This property is incredibly useful for solving equations that have been factored, as it allows us to break down a complex equation into simpler ones.
In our case, we've factored the quadratic equation x² - 7x + 10 = 0 into (x - 2)(x - 5) = 0. We now have a product of two factors, (x - 2) and (x - 5), that equals zero. According to the zero-product property, this means that either (x - 2) = 0 or (x - 5) = 0 (or both). This step is crucial because it transforms a single quadratic equation into two linear equations, which are much easier to solve.
Now, we simply need to solve each of these linear equations separately. For the first equation, (x - 2) = 0, we add 2 to both sides to isolate x, which gives us x = 2. For the second equation, (x - 5) = 0, we add 5 to both sides to isolate x, which gives us x = 5. So, we have found two potential solutions for our original quadratic equation: x = 2 and x = 5.
The zero-product property provides a direct and straightforward way to find the solutions once the equation is factored. It's like a bridge that connects the factored form of the equation to its solutions. Without this property, we wouldn't be able to easily determine the values of x that make the equation true. By applying this property, we've successfully narrowed down the possible solutions and are now ready to confirm that these values indeed satisfy the original equation.
Finding the Solutions
Now that we've applied the zero-product property to our factored equation (x - 2)(x - 5) = 0, we've arrived at two simple linear equations: x - 2 = 0 and x - 5 = 0. These equations are straightforward to solve. For the first equation, x - 2 = 0, we can isolate x by adding 2 to both sides of the equation. This gives us x = 2. For the second equation, x - 5 = 0, we can isolate x by adding 5 to both sides, resulting in x = 5.
So, we've found two potential solutions for the quadratic equation x² - 7x + 10 = 0: x = 2 and x = 5. These values represent the roots of the equation, which are the values of x that make the equation true. To confirm that these are indeed the solutions, we can substitute each value back into the original equation and check if it holds true. This is a crucial step in the problem-solving process, as it verifies the accuracy of our calculations and ensures that we haven't made any mistakes along the way.
Let's first substitute x = 2 into the original equation: (2)² - 7(2) + 10 = 4 - 14 + 10 = 0. The equation holds true, so x = 2 is indeed a solution. Next, let's substitute x = 5 into the original equation: (5)² - 7(5) + 10 = 25 - 35 + 10 = 0. Again, the equation holds true, confirming that x = 5 is also a solution. Therefore, we have successfully identified both solutions to the quadratic equation.
These solutions, x = 2 and x = 5, are the values where the parabola represented by the quadratic equation intersects the x-axis. In other words, they are the x-intercepts of the graph of the equation. By finding these solutions, we've completely solved the quadratic equation using the factoring method and the zero-product property. Now, we can confidently express the solution set.
The Solution Set
We've found that the solutions to the equation x² - 7x + 10 = 0 are x = 2 and x = 5. The solution set is simply a way of presenting these solutions in a clear and concise manner. It's a set that contains all the values of x that satisfy the equation. In mathematics, a set is typically denoted by curly braces { }.
Therefore, the solution set for our equation is {2, 5}. This means that if we substitute either 2 or 5 for x in the original equation, the equation will be true. No other values will satisfy the equation. The solution set is the final answer to the problem, and it represents the complete set of solutions.
Looking at the multiple-choice options given, we can see that option (A) {2, 5} matches our solution set. So, that's the correct answer! We've successfully solved the quadratic equation by factoring, applying the zero-product property, and verifying our solutions.
It's always a good idea to double-check your work, especially in math problems. We've already verified our solutions by plugging them back into the original equation, but let's just quickly recap the steps we took. We factored the quadratic expression, applied the zero-product property, solved the resulting linear equations, and presented the solutions in a set. Each step was logical and based on fundamental algebraic principles. By understanding these principles and practicing problem-solving techniques, we can tackle even more complex quadratic equations with confidence.
Conclusion
So, guys, we've successfully navigated through solving the quadratic equation x² - 7x + 10 = 0! We used the factoring method, the zero-product property, and a bit of logical thinking to arrive at the solution set {2, 5}. Remember, quadratic equations might seem daunting at first, but breaking them down into manageable steps makes them much easier to handle. Keep practicing, and you'll become a pro at solving these in no time! If you have any questions or want to explore other methods for solving quadratic equations, feel free to ask. Keep up the great work!
