Solving Quadratic Equations: A Step-by-Step Guide

Hey guys! Let's dive into a classic math problem that involves quadratic equations. We're going to break down how to solve this kind of problem step by step, making it super easy to understand. So, the question goes like this: We know that x₁ and x₂ are the roots of the quadratic equation 2x² - 3x + 4 = 0. A new quadratic equation has roots of 2x₁ and 2x₂, and it's expressed as x² - ax + b = 0. The goal? To figure out which statement is correct regarding a and b. Let's get started!

Understanding the Basics of Quadratic Equations

Alright, before we jump into the problem, let's quickly recap what a quadratic equation is all about. A quadratic equation is basically an equation that looks like this: ax² + bx + c = 0, where a, b, and c are constants, and a is not zero. The solutions to a quadratic equation are called roots, and they're the values of x that make the equation true. There are several ways to find these roots, including factoring, completing the square, and using the quadratic formula. In our case, we're not going to solve for x₁ and x₂ directly but rather use their properties to find the new equation. This is a common trick in math problems, so let's pay attention!

When dealing with quadratic equations, there are some handy relationships between the coefficients and the roots. For a quadratic equation ax² + bx + c = 0, the sum of the roots (x₁ + x₂) is equal to -b/ a, and the product of the roots (x₁ * x₂) is equal to c/ a. These relationships are super useful because they allow us to find the sum and product of the roots without actually solving the equation for its roots. Remember these, because they'll save you time and effort on many problems. They are a must-know for anyone trying to get the hang of quadratic equations, so make sure you understand them!

Finding the Sum and Product of the Original Roots

Okay, back to our problem. We know the original equation is 2x² - 3x + 4 = 0. Let's use the relationships we just talked about to find the sum and product of its roots, x₁ and x₂. Here, a = 2, b = -3, and c = 4.

First, let's find the sum of the roots: x₁ + x₂ = -b/ a = -(-3) / 2 = 3/2.

Next, let's find the product of the roots: x₁ * x₂ = c/ a = 4 / 2 = 2.

See? We got these values without solving for x₁ and x₂ directly. That's the beauty of using these relationships! Keep in mind how important this step is. This allows you to work with the properties of the roots instead of the roots themselves. Pretty cool, huh? The ability to do this will help make other problems a breeze.

Forming the New Quadratic Equation

Now, we need to create a new quadratic equation whose roots are 2x₁ and 2x₂. We know that a general quadratic equation can be written as x² - (sum of roots)x + (product of roots) = 0. So, we need to find the sum and product of the new roots.

Let's calculate the sum of the new roots: 2x₁ + 2x₂ = 2(x₁ + x₂) = 2 * (3/2) = 3.

Now, let's calculate the product of the new roots: (2x₁) * (2x₂) = 4(x₁ * x₂) = 4 * 2 = 8.

So, the new quadratic equation is x² - 3x + 8 = 0. Comparing this with the given form x² - ax + b = 0, we can see that a = 3 and b = 8.

Analyzing the Results

We've found that a = 3 and b = 8. Now let's analyze the statements given in the question to determine the correct one.

• a. a is a non-prime number and b is a multiple of 3: Since a = 3, which is a prime number, this statement is incorrect. Also, b = 8 is not a multiple of 3.
• b. a is a prime number and b is a multiple of 3: We know that a = 3, which is a prime number, but b = 8, which is not a multiple of 3. So, this statement is also incorrect.

It seems that there was a mix-up with the possible answers. The most accurate observation from our calculations is that a (which is 3) is a prime number, and b (which is 8) is not a multiple of 3. But this does not precisely fit with any of the two options. The question must have been designed for us to focus on understanding the quadratic formula properties.

Conclusion and Key Takeaways

Alright, guys, we've walked through solving a quadratic equation problem, where the core skill is understanding relationships between the roots and coefficients. We found the sum and product of the original roots using the formulas -b/ a and c/ a. Then, we used those to find the new quadratic equation, and therefore the values of a and b. The trick is to be able to transform the original equation. Knowing these relationships saves you a lot of time and makes solving quadratic equations much easier!

Key Takeaways

• Always remember the formulas for the sum and product of roots: x₁ + x₂ = -b/ a and x₁ * x₂ = c/ a.
• Understand how to form a new quadratic equation given the roots.
• Pay close attention to the details of the problem and how to manipulate equations and apply formulas.
• Practice and review these concepts to get even better. Keep in mind that understanding and practicing this type of problem is critical for other topics as well.

Keep practicing, and you'll become a pro at solving these types of problems in no time! Remember, math is all about practice. The more you work with these equations, the more comfortable you'll become, and the faster you'll solve them.