The Product of Solutions in a Quadratic Equation

Oct 13, 2025 by windailys.com 42 views

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Introduction

In the given quadratic equation $$57x^2 + (57b + a)x + ab = 0$$, we are asked to find the value of the constant $k$ in the product of the solutions to the equation, which is given as $ka b$ . To solve this problem, we need to use the properties of quadratic equations and the relationship between the coefficients of the equation and the product of its solutions.

The Product of Solutions in a Quadratic Equation

The product of the solutions to a quadratic equation of the form $ax^2 + bx + c = 0$ is given by the formula $\frac{c}{a}$ . In our given equation, the product of the solutions is $ka b$ , where $k$ is a constant. We can use this information to find the value of $k$ .

Using Vieta's Formulas

Vieta's formulas state that for a quadratic equation of the form $ax^2 + bx + c = 0$ , the sum of the solutions is $-\frac{b}{a}$ and the product of the solutions is $\frac{c}{a}$ . In our given equation, the product of the solutions is $ka b$ , so we can set up the equation $\frac{ab}{57} = ka b$ .

Solving for $k$

To solve for $k$ , we can start by multiplying both sides of the equation by 57 to get rid of the fraction. This gives us $ab = 57ka b$ . We can then divide both sides of the equation by $ab$ to get $1 = 57k$ .

Finding the Value of $k$

To find the value of $k$ , we can divide both sides of the equation by 57. This gives us $k = \frac{1}{57}$ .

Conclusion

In this problem, we were asked to find the value of the constant $k$ in the product of the solutions to the given quadratic equation. We used Vieta's formulas to find the product of the solutions and then solved for $k$ . The final answer is $k = \frac{1}{57}$ .

Example Use Case

This problem can be used to illustrate the concept of Vieta's formulas and the relationship between the coefficients of a quadratic equation and the product of its solutions. It can also be used to practice solving quadratic equations and finding the product of their solutions.

Step-by-Step Solution

Write down the given quadratic equation: $$57x^2 + (57b + a)x + ab = 0$$
Use Vieta's formulas to find the product of the solutions: $\frac{ab}{57} = ka b$
Multiply both sides of the equation by 57 to get rid of the fraction: $ab = 57ka b$
Divide both sides of the equation by $ab$ to get $1 = 57k$
Divide both sides of the equation by 57 to find the value of $k$ : $k = \frac{1}{57}$

Key Takeaways

The product of the solutions to a quadratic equation is given by the formula $\frac{c}{a}$ .
Vieta's formulas state that for a quadratic equation of the form $ax^2 + bx + c = 0$ , the sum of the solutions is $-\frac{b}{a}$ and the product of the solutions is $\frac{c}{a}$ .
To find the value of $k$ in the product of the solutions to a quadratic equation, we can use Vieta's formulas and solve for $k$ .

Frequently Asked Questions

What is the product of the solutions to a quadratic equation?
- The product of the solutions to a quadratic equation is given by the formula $\frac{c}{a}$ .
How do we find the value of $k$ $k$ in the product of the solutions to a quadratic equation?
- We can use Vieta's formulas and solve for $k$ .
What is Vieta's formulas?
- Vieta's formulas state that for a quadratic equation of the form $ax^2 + bx + c = 0$ , the sum of the solutions is $-\frac{b}{a}$ and the product of the solutions is $\frac{c}{a}$ .

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Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding their solutions is crucial for various applications in science, engineering, and other fields. In this article, we will provide a comprehensive Q&A guide on quadratic equation solutions, covering various topics and concepts.

Q&A: Quadratic Equation Solutions

Q1: What is a quadratic equation?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. It is typically written in the form ax^2 + bx + c = 0, where a, b, and c are constants.

Q2: How do I solve a quadratic equation?

There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and completing the square. The most common method is using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.

Q3: What is the quadratic formula?

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by: x = (-b ± √(b^2 - 4ac)) / 2a.

Q4: What is the difference between the sum and product of roots?

The sum of the roots of a quadratic equation is given by -b/a, while the product of the roots is given by c/a.

Q5: How do I find the sum and product of roots?

To find the sum and product of roots, you can use Vieta's formulas. The sum of the roots is -b/a, and the product of the roots is c/a.

Q6: What is Vieta's formulas?

Vieta's formulas are a set of mathematical formulas that relate the coefficients of a polynomial to the sums and products of its roots.

Q7: How do I use Vieta's formulas?

To use Vieta's formulas, you need to identify the coefficients of the polynomial and then apply the formulas to find the sum and product of the roots.

Q8: What is the relationship between the coefficients and the roots?

The coefficients of a polynomial are related to the sums and products of its roots through Vieta's formulas.

Q9: How do I find the value of k in the product of the solutions?

To find the value of k in the product of the solutions, you can use the formula: k = c/a.

Q10: What is the significance of the product of the solutions?

The product of the solutions is an important concept in mathematics, as it provides information about the roots of a polynomial.

Conclusion

In this article, we have provided a comprehensive Q&A guide on quadratic equation solutions, covering various topics and concepts. We hope that this guide has been helpful in understanding the solutions to quadratic equations and their applications.

Example Use Cases

Science: Quadratic equations are used to model the motion of objects under the influence of gravity, such as the trajectory of a projectile.
Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
Computer Science: Quadratic equations are used in algorithms and data structures, such as sorting and searching.

Step-by-Step Solution

Write down the quadratic equation: ax^2 + bx + c = 0
Use the quadratic formula to find the solutions: x = (-b ± √(b^2 - 4ac)) / 2a
Identify the coefficients of the polynomial: a, b, and c
Use Vieta's formulas to find the sum and product of the roots: sum = -b/a and product = c/a
Find the value of k in the product of the solutions: k = c/a

Key Takeaways

Quadratic equations are a fundamental concept in mathematics.
The quadratic formula provides the solutions to a quadratic equation.
Vieta's formulas relate the coefficients of a polynomial to the sums and products of its roots.
The product of the solutions is an important concept in mathematics.

Frequently Asked Questions

What is a quadratic equation?
- A quadratic equation is a polynomial equation of degree two.
How do I solve a quadratic equation?
- There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and completing the square.
What is the quadratic formula?
- The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation.
What is Vieta's formulas?
- Vieta's formulas are a set of mathematical formulas that relate the coefficients of a polynomial to the sums and products of its roots.