Solving For A And B When A + B = 10 And Ab = 10

Aug 3, 2025 by Pedro Alvarez 48 views

Hey guys! Let's dive into this interesting math problem where we need to find the values of a and b given two equations: a + b = 10 and ab = 10. It might seem tricky at first, but we'll break it down step by step. We will explore different methods to solve this problem, ensuring you understand the underlying concepts and can tackle similar questions with ease. Forget just picking an answer; we're going to learn how to solve it, making sure you're confident in your math skills. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving, let’s make sure we really get what the problem is asking. We have two key pieces of information here, which are presented as equations:

a + b = 10: This tells us that if we add the two numbers, a and b, we should get 10. Think of it like this: a and b are two parts that, together, make up 10. The sum of a and b is 10.
ab = 10: This means that when we multiply a and b, we also get 10. In other words, the product of a and b is 10.

Our mission is to find out what specific numbers a and b are, that satisfy both these conditions at the same time. This type of problem, where we have multiple equations and multiple unknowns, often requires us to use a bit of algebraic manipulation and some logical thinking.

Now, let's take a look at the options given to us. The options provided are:

A) a = 2, b = 5
B) a = 5, b = 5
C) a = 1, b = 9
D) a = 2, b = 8

One way we could tackle this is by testing each option to see if it fits both equations. However, that might take a bit of time and doesn't really teach us how to solve this kind of problem generally. Instead, let's explore some algebraic methods that will not only help us solve this particular problem but also equip us with the skills to solve similar problems in the future. We'll dive into techniques like substitution and the quadratic formula. Understanding these methods will give you a more robust understanding and the ability to approach these mathematical challenges confidently. We will first try to solve the problem using algebraic manipulations and then check if any of the given options match our solution. This way, we're not just guessing and checking; we're actually solving the problem.

Methods to Solve

Okay, so we know the problem, and we have some potential answers. But let's solve this thing properly! We're going to explore a couple of different methods to find the values of a and b. These methods aren't just useful for this question; they're key skills in algebra that you'll use again and again.

Method 1: Substitution

Substitution is a neat trick where we rearrange one equation to isolate one variable and then plug that expression into the other equation. Here’s how it works for our problem:

Rearrange the first equation: We'll start with a + b = 10. Let's solve for a. We can subtract b from both sides to get: a = 10 - b.
Substitute into the second equation: Now we know that a is the same as 10 - b. So, wherever we see a in the second equation (ab = 10), we can replace it with (10 - b). This gives us: (10 - b) b = 10.
Simplify and rearrange: Let’s expand and rearrange this equation. Multiplying out the brackets gives us 10b - b² = 10. If we bring everything to one side, we get a quadratic equation: b² - 10b + 10 = 0. Guys, recognizing this as a quadratic equation is a crucial step! It means we have a b² term, a b term, and a constant. This form allows us to use methods specifically designed for solving quadratic equations.

Method 2: Using the Quadratic Formula

So, we’ve got a quadratic equation: b² - 10b + 10 = 0. Now, how do we solve it? This is where the quadratic formula comes to the rescue! The quadratic formula is a powerful tool that gives us the solutions for any quadratic equation in the form ax² + bx + c = 0. It looks a bit scary, but it’s really just a plug-and-chug formula:

x = (-b ± √(b² - 4ac)) / (2a)

In our equation, b² - 10b + 10 = 0, we can identify a, b, and c as follows:

a = 1 (the coefficient of b²)
b = -10 (the coefficient of b)
c = 10 (the constant term)

Now, let's plug these values into the quadratic formula. You might be thinking, "Whoa, this looks intense!" But trust me, it's just about careful substitution and following the order of operations. We're breaking it down into manageable steps so you can see exactly how it works. Remember, the key to mastering math is not just memorizing formulas but understanding how to apply them. So, let’s get this done, and you'll feel like a quadratic formula pro!

b = (-(-10) ± √((-10)² - 4 * 1 * 10)) / (2 * 1)

Let's simplify this step by step:

b = (10 ± √(100 - 40)) / 2

b = (10 ± √60) / 2

b = (10 ± 2√15) / 2

Now, we can simplify further by dividing both terms in the numerator by 2:

b = 5 ± √15

So, we have two possible values for b:

b₁ = 5 + √15
b₂ = 5 - √15

Finding a

We've found two possible values for b. Now we need to find the corresponding values for a. Remember our equation a = 10 - b? We'll use that to find a for each value of b.

For b₁ = 5 + √15:

a₁ = 10 - (5 + √15)

a₁ = 10 - 5 - √15

a₁ = 5 - √15

For b₂ = 5 - √15:

a₂ = 10 - (5 - √15)

a₂ = 10 - 5 + √15

a₂ = 5 + √15

So, we have two pairs of solutions:

a₁ = 5 - √15, b₁ = 5 + √15
a₂ = 5 + √15, b₂ = 5 - √15

Notice that the values of a and b are just switched in the two solutions. This makes sense because the original equations are symmetrical in a and b. Swapping a and b doesn't change the equations.

Checking the Options

Okay, we've done the hard work and solved for a and b. Now, let's take a look at those options again and see if any of them match our solutions. Our solutions are:

a = 5 - √15, b = 5 + √15
a = 5 + √15, b = 5 - √15

Let’s revisit the options:

A) a = 2, b = 5
B) a = 5, b = 5
C) a = 1, b = 9
D) a = 2, b = 8

Looking at these, we can see that none of the options perfectly match our solutions. Our solutions involve the square root of 15, which is an irrational number, meaning it cannot be expressed as a simple fraction or a terminating decimal. The options provided are all whole numbers.

This tells us something important: The problem, as stated with those specific answer choices, doesn't have a solution within the given options. It’s crucial to recognize this! In math, it's just as important to know when a problem has no solution within a given set of constraints as it is to find a solution. This could be a situation where the options provided are incorrect, or the problem was designed to test your ability to recognize unsolvable scenarios with the provided constraints.

Justification and Explanation

So, to wrap things up, we've gone through a detailed process to solve for a and b when a + b = 10 and ab = 10. We used substitution to get a quadratic equation, then we applied the quadratic formula to find the exact values of b. From there, we easily calculated the corresponding values of a. What we discovered is that the correct values of a and b involve √15, which means they aren't whole numbers. None of the given options match these solutions.

Therefore, the correct answer is that none of the options (A, B, C, or D) are correct for the equations a + b = 10 and ab = 10. The problem highlights the importance of not just blindly picking an answer but actually solving the problem and understanding the nature of the solutions. In this case, the solutions are irrational numbers, which are not represented in the provided options.

This exercise is a great example of how math isn’t just about getting the "right" answer; it’s about the process, the understanding, and the ability to recognize when something doesn't quite fit. You guys tackled this like pros, and hopefully, you've added some valuable problem-solving skills to your toolkit!