How to Find Solutions to an Equation

Oct 13, 2025 by wbrum.org 129 views

Introduction

In mathematics, an equation is a statement that expresses the equality of two mathematical expressions. Solving an equation involves finding the values of the variables that make the equation true. In this article, we will explore which equation has solutions of -2 and 3.

What are Solutions to an Equation?

A solution to an equation is a value of the variable that makes the equation true. In other words, it is a value that, when substituted into the equation, makes the equation equal to zero. For example, if we have the equation x + 2 = 5, then the solution to the equation is x = 3, because when we substitute x = 3 into the equation, we get 3 + 2 = 5, which is true.

How to Find Solutions to an Equation

There are several methods to find solutions to an equation, including factoring, using the quadratic formula, and graphing. In this article, we will use the factoring method to find the solutions to the given equations.

The Given Equations

We are given four equations:

A. $y = x^2 + x - 6$ B. $y = x^2 - x - 6$ C. $y = x^2 + 5x + 6$ D. $y = x^2 - 5x + 6$

We need to find which of these equations has solutions of -2 and 3.

Factoring the Equations

To find the solutions to the equations, we need to factor them. Factoring an equation involves expressing it as a product of two or more binomials.

Equation A: $y = x^2 + x - 6$

To factor this equation, we need to find two numbers whose product is -6 and whose sum is 1. These numbers are 2 and -3, so we can factor the equation as:

$y = (x + 2)(x - 3)$

Equation B: $y = x^2 - x - 6$

To factor this equation, we need to find two numbers whose product is -6 and whose sum is -1. These numbers are -2 and 3, so we can factor the equation as:

$y = (x - 2)(x + 3)$

Equation C: $y = x^2 + 5x + 6$

To factor this equation, we need to find two numbers whose product is 6 and whose sum is 5. These numbers are 2 and 3, so we can factor the equation as:

$y = (x + 2)(x + 3)$

Equation D: $y = x^2 - 5x + 6$

To factor this equation, we need to find two numbers whose product is 6 and whose sum is -5. These numbers are -2 and -3, so we can factor the equation as:

$y = (x - 2)(x - 3)$

Analyzing the Factored Equations

Now that we have factored the equations, we can analyze them to see which one has solutions of -2 and 3.

Equation A: $y = (x + 2)(x - 3)$

This equation has solutions of x = -2 and x = 3, because when we substitute x = -2 into the equation, we get (-2 + 2)(-2 - 3) = (0)(-5) = 0, which is true. Similarly, when we substitute x = 3 into the equation, we get (3 + 2)(3 - 3) = (5)(0) = 0, which is also true.

Equation B: $y = (x - 2)(x + 3)$

This equation does not have solutions of x = -2 and x = 3, because when we substitute x = -2 into the equation, we get (-2 - 2)(-2 + 3) = (-4)(1) = -4, which is not equal to zero. Similarly, when we substitute x = 3 into the equation, we get (3 - 2)(3 + 3) = (1)(6) = 6, which is also not equal to zero.

Equation C: $y = (x + 2)(x + 3)$

This equation does not have solutions of x = -2 and x = 3, because when we substitute x = -2 into the equation, we get (-2 + 2)(-2 + 3) = (0)(1) = 0, which is not equal to zero. Similarly, when we substitute x = 3 into the equation, we get (3 + 2)(3 + 3) = (5)(6) = 30, which is also not equal to zero.

Equation D: $y = (x - 2)(x - 3)$

This equation does not have solutions of x = -2 and x = 3, because when we substitute x = -2 into the equation, we get (-2 - 2)(-2 - 3) = (-4)(-5) = 20, which is not equal to zero. Similarly, when we substitute x = 3 into the equation, we get (3 - 2)(3 - 3) = (1)(0) = 0, which is also not equal to zero.

Conclusion

In conclusion, the equation that has solutions of -2 and 3 is Equation A: $y = x^2 + x - 6$ . This equation can be factored as $y = (x + 2)(x - 3)$ , and when we substitute x = -2 and x = 3 into the equation, we get true statements.

Final Answer

Introduction

In our previous article, we explored which equation has solutions of -2 and 3. We analyzed four different equations and found that Equation A: $y = x^2 + x - 6$ is the one that has solutions of -2 and 3. In this article, we will answer some frequently asked questions about the topic.

Q: What is the significance of solutions to an equation?

A: Solutions to an equation are values of the variable that make the equation true. In other words, they are values that, when substituted into the equation, make the equation equal to zero. Solutions are important in mathematics because they help us understand the behavior of the equation and make predictions about the real-world phenomena that the equation models.

Q: How do you find solutions to an equation?

A: There are several methods to find solutions to an equation, including factoring, using the quadratic formula, and graphing. In our previous article, we used the factoring method to find the solutions to the given equations.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to find the solutions to a quadratic equation. The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where a, b, and c are the coefficients of the quadratic equation.

Q: How do you use the quadratic formula to find solutions to an equation?

A: To use the quadratic formula to find solutions to an equation, you need to identify the coefficients a, b, and c of the equation. Then, you plug these values into the quadratic formula and simplify the expression to find the solutions.

Q: What is the difference between a solution and a root of an equation?

A: A solution to an equation is a value of the variable that makes the equation true. A root of an equation is a value of the variable that makes the equation equal to zero. In other words, a solution is a value that satisfies the equation, while a root is a value that makes the equation equal to zero.

Q: Can an equation have multiple solutions?

A: Yes, an equation can have multiple solutions. In fact, most equations have multiple solutions, and the number of solutions depends on the degree of the equation. For example, a linear equation has one solution, while a quadratic equation has two solutions.

Q: Can an equation have no solutions?

A: Yes, an equation can have no solutions. This happens when the equation is inconsistent, meaning that it is impossible to find a value of the variable that makes the equation true.

Q: What is the importance of understanding solutions to an equation?

A: Understanding solutions to an equation is important because it helps us understand the behavior of the equation and make predictions about the real-world phenomena that the equation models. Solutions to an equation can also be used to solve problems in various fields, such as physics, engineering, and economics.

Conclusion

In conclusion, understanding solutions to an equation is crucial in mathematics and has many practical applications. We hope that this Q&A article has helped you understand the concept of solutions to an equation and how to find them using different methods.

Final Answer

The final answer is Equation A: $y = x^2 + x - 6$ .