Solve Quadratic Equation: Find Root & Coefficient

Nov 22, 2025 by Andrew McMorgan 50 views

Solving Quadratic Equations: Finding the Other Root and Coefficient

Hey Plastik Magazine readers! Let's dive into a cool math problem today. We're going to tackle a quadratic equation, and it's going to be fun, I promise. We'll explore how to find the missing root and a mystery coefficient when we know one root already. Get your thinking caps on, and let's get started!

Understanding Quadratic Equations

Before we jump into the problem, let's quickly recap what a quadratic equation is. A quadratic equation is basically a polynomial equation of the second degree. You'll usually see it written in the standard form: $ax^2 + bx + c = 0$ , where a, b, and c are coefficients, and x is our variable. The solutions to this equation are called roots, and they're the values of x that make the equation true.

Finding the roots is a pretty common task in algebra. There are several methods to do it, such as factoring, using the quadratic formula, or even completing the square. But in our case, we have a little head start – we already know one of the roots. This makes our job much easier!

Key Concepts for Solving: Understanding the relationship between roots and coefficients is essential for solving quadratic equations. For a quadratic equation in the form of $ax^2 + bx + c = 0$ , if the roots are $x_1$ and $x_2$ , then:

Sum of the roots: $x_1 + x_2 = -b/a$
Product of the roots: $x_1 * x_2 = c/a$

These relationships are derived from Vieta's formulas, a powerful tool in algebra that simplifies the process of finding roots and coefficients. By utilizing these formulas, we can efficiently solve for unknowns without resorting to more complex methods like the quadratic formula, especially when some information is already provided.

When solving quadratic equations, always double-check your solutions by substituting them back into the original equation. This ensures that the roots you've found are accurate and satisfy the equation. Additionally, be mindful of the context of the problem; sometimes, certain solutions may not be viable (e.g., negative lengths in a geometric problem).

The Problem: Unveiling the Root and the Coefficient

Okay, here's the challenge we're facing: One root of the equation $5x^2 + bx + 24 = 0$ is 8. Our mission, should we choose to accept it (and we totally do!), is to find the other root and the value of the coefficient b. So, we know part of the puzzle, but we need to fill in the missing pieces. How cool is that?

Let's break down the info we have. We have a quadratic equation, $5x^2 + bx + 24 = 0$ . We know one root, let's call it $x_1$ , is 8. We need to find the other root, $x_2$ , and the value of b. This is where our understanding of the relationships between roots and coefficients comes into play.

Now, remember those formulas we talked about earlier? The sum and product of roots? They're going to be our best friends here. We know that the product of the roots is $c/a$ , and the sum is $-b/a$ . In our equation, a is 5, c is 24, and b is, well, what we're trying to find! So, let's put these concepts to work and solve this mystery.

Applying the Concepts: The problem presents a quadratic equation, $5x^2 + bx + 24 = 0$ , with one known root of 8. To solve for the other root and the coefficient b, we can use the relationships between the roots and coefficients of a quadratic equation. These relationships, derived from Vieta's formulas, state that for a quadratic equation of the form $ax^2 + bx + c = 0$ , the sum of the roots is equal to $-b/a$ , and the product of the roots is equal to $c/a$ .

In this specific equation, $a = 5$ and $c = 24$ . Since we know one root ( $x_1 = 8$ ), we can use the product of roots formula to find the other root ( $x_2$ ). The formula is $x_1 * x_2 = c/a$ . Substituting the known values gives us $8 * x_2 = 24/5$ . Solving for $x_2$ will give us the value of the second root.

Once we have both roots, we can use the sum of roots formula to find the value of b. The formula is $x_1 + x_2 = -b/a$ . We know $x_1$ , we will have calculated $x_2$ , and we know $a$ . Substituting these values into the formula will allow us to solve for b. This step is crucial in completing the solution, as it connects the roots back to the original equation's coefficient.

This approach provides a systematic way to solve for both the unknown root and the coefficient, leveraging the fundamental relationships within quadratic equations. It's a demonstration of how theoretical concepts can be directly applied to solve practical problems, making it an essential skill for anyone studying algebra or related fields.

Cracking the Code: Step-by-Step Solution

Alright, let's put on our detective hats and solve this thing step by step. First, we'll use the product of roots formula to find the other root. Then, we'll use the sum of roots formula to uncover the value of b. Ready? Let's do it!

Finding the Other Root ( $x_2$ ): Remember, the product of the roots is $c/a$ . In our case, that's $24/5$ . We know one root ( $x_1$ ) is 8, so we can set up the equation: $8 * x_2 = 24/5$ . To solve for $x_2$ , we divide both sides by 8: $x_2 = (24/5) / 8$ . Simplify that, and we get $x_2 = 3/5$ . Boom! We found the other root. It's 3/5.
Unveiling the Coefficient b: Now that we have both roots, we can use the sum of roots formula to find b. The sum of the roots is $-b/a$ . We know $x_1 = 8$ , $x_2 = 3/5$ , and $a = 5$ . So, $8 + 3/5 = -b/5$ . Let's simplify the left side: $8 + 3/5 = 43/5$ . Now we have $43/5 = -b/5$ . To solve for b, we multiply both sides by -5: $b = -43$ . There you have it! The coefficient b is -43.

Detailed Steps and Calculations: Let’s break down the solution into precise steps to ensure clarity and accuracy. This detailed approach not only solves the problem but also reinforces the understanding of the underlying principles of quadratic equations.

Step 1: Identify Known Values: First, we identify the values given in the problem. We have the quadratic equation $5x^2 + bx + 24 = 0$ , one root $x_1 = 8$ , and we need to find the other root $x_2$ and the coefficient b.
Step 2: Apply the Product of Roots Formula: The product of the roots for a quadratic equation $ax^2 + bx + c = 0$ is given by $x_1 * x_2 = c/a$ . In our case, $a = 5$ and $c = 24$ , so the product of the roots is $24/5$ . We know one root is 8, so we set up the equation $8 * x_2 = 24/5$ .
Step 3: Solve for the Second Root ( $x_2$ ): To find $x_2$ , we divide both sides of the equation by 8: $x_2 = (24/5) / 8$ . Simplifying this, we get $x_2 = 24 / (5 * 8) = 24 / 40$ . Further simplification by dividing both numerator and denominator by 8 gives us $x_2 = 3/5$ . Therefore, the second root is $3/5$ .
Step 4: Apply the Sum of Roots Formula: The sum of the roots is given by $x_1 + x_2 = -b/a$ . We know $x_1 = 8$ , $x_2 = 3/5$ , and $a = 5$ . So, we set up the equation $8 + 3/5 = -b/5$ .
Step 5: Simplify the Sum of Roots: To add the roots, we convert 8 to a fraction with a denominator of 5, giving us $40/5$ . Thus, the equation becomes $40/5 + 3/5 = -b/5$ , which simplifies to $43/5 = -b/5$ .
Step 6: Solve for the Coefficient b: To find b, we multiply both sides of the equation by -5. This gives us $-5 * (43/5) = -5 * (-b/5)$ , which simplifies to $-43 = b$ . Therefore, the coefficient b is -43.
Step 7: Verify the Solution: To ensure our solution is correct, we can substitute both roots back into the original equation and check if it holds true. Additionally, we can use the quadratic formula to find the roots of the equation $5x^2 - 43x + 24 = 0$ and verify that they match our calculated roots. This step is crucial for confirming the accuracy of our solution.

By following these detailed steps, we not only solve the problem but also reinforce the understanding of how to apply Vieta’s formulas in solving quadratic equations. This methodical approach is essential for mastering algebraic concepts and ensuring accurate problem-solving.

The Big Reveal: Answers Unveiled!

So, after all that math-sleuthing, we've arrived at our answers. The other root of the equation is 3/5, and the value of the coefficient b is -43. How awesome is that? We took a puzzle, used our knowledge of quadratic equations, and cracked the code. You guys are math wizards!

Results Summary: To summarize, we found that the other root of the equation $5x^2 + bx + 24 = 0$ is $x_2 = 3/5$ , and the coefficient b is -43. These values satisfy the given conditions and complete the solution to the problem.

This exercise highlights the practical application of algebraic principles in solving real-world problems. By understanding the relationships between roots and coefficients, we can efficiently tackle quadratic equations and similar challenges. It also underscores the importance of meticulous calculations and verification in mathematical problem-solving, ensuring accuracy and confidence in our results.

Why This Matters: Real-World Connections

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