Solving the Correct Equation

Oct 13, 2025 by recoveryshake.com 41 views

Understanding the Problem

When dealing with equations involving products of two numbers, it's essential to identify the correct equation to solve for the value of the smaller number. In this case, we're given the product of two numbers, $$x$$ and $$x+3$$, which equals 550. Our goal is to determine the correct equation that could be solved to find the value of the smaller number, $$x$$.

Analyzing the Options

Let's analyze each option to determine which one represents the correct equation to solve for $$x$$.

Option A: $$x^2 + 3 = 550$$

This option suggests that the equation is a quadratic equation, where $$x^2$$ is the product of $$x$$ and $$x+3$$. However, this equation is not entirely accurate, as it doesn't account for the $$+3$$ term in the product. To correctly represent the product of $$x$$ and $$x+3$$, we need to multiply the two terms together, resulting in $$x(x+3)$$.

Option B: $$3x^2 = 500$$

This option suggests that the equation is a quadratic equation, where $$3x^2$$ represents the product of $$x$$ and $$x+3$$. However, this equation is not entirely accurate, as it doesn't account for the $$+3$$ term in the product. To correctly represent the product of $$x$$ and $$x+3$$, we need to multiply the two terms together, resulting in $$x(x+3)$$.

Option C: $$x(x+3) = 550$$

This option correctly represents the product of $$x$$ and $$x+3$$, which equals 550. By multiplying the two terms together, we get $$x(x+3) = x^2 + 3x$$, which is equivalent to $$x^2 + 3x = 550$$. This equation can be rewritten as $$x^2 + 3x - 550 = 0$$, which is a quadratic equation that can be solved to find the value of $$x$$.

Solving the Correct Equation

To solve the correct equation, $$x^2 + 3x - 550 = 0$$, we can use the quadratic formula:

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

In this case, $$a = 1$$, $$b = 3$$, and $$c = -550$$. Plugging these values into the quadratic formula, we get:

x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-550)}}{2(1)}

Simplifying the expression under the square root, we get:

x = \frac{-3 \pm \sqrt{9 + 2200}}{2}

x = \frac{-3 \pm \sqrt{2209}}{2}

x = \frac{-3 \pm 47}{2}

This gives us two possible solutions for $$x$$:

x = \frac{-3 + 47}{2} = 22

x = \frac{-3 - 47}{2} = -25

Since we're looking for the smaller number, $$x$$, the correct solution is $$x = -25$$.

Conclusion

In conclusion, the correct equation to solve for the value of the smaller number, $$x$$, is $$x^2 + 3x - 550 = 0$$. This equation can be solved using the quadratic formula, resulting in two possible solutions for $$x$$. The correct solution is $$x = -25$$, which represents the smaller number.

Frequently Asked Questions

What is the product of two numbers, $$x$$ and $$x+3$$? The product of two numbers, $$x$$ and $$x+3$$, is $$x(x+3)$$.
How do I solve the correct equation to find the value of the smaller number, $$x$$? To solve the correct equation, $$x^2 + 3x - 550 = 0$$, you can use the quadratic formula.
What is the correct solution for $$x$$? The correct solution for $$x$$ is $$x = -25$$.

Additional Resources

Quadratic Formula: A quadratic equation in the form $$ax^2 + bx + c = 0$$ can be solved using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$.
Solving Quadratic Equations: There are several methods for solving quadratic equations, including factoring, the quadratic formula, and graphing.
Algebra: Algebra is a branch of mathematics that deals with the study of variables and their relationships. It includes topics such as linear equations, quadratic equations, and functions.

Introduction

In our previous article, we discussed how to select the correct answer when solving a quadratic equation involving the product of two numbers. We also solved the correct equation to find the value of the smaller number, $$x$$. In this article, we'll answer some frequently asked questions (FAQs) related to solving quadratic equations and finding the smaller number.

Q&A

Q: What is the product of two numbers, $$x$$ and $$x+3$$?

A: The product of two numbers, $$x$$ and $$x+3$$, is $$x(x+3)$$.

Q: How do I solve the correct equation to find the value of the smaller number, $$x$$?

A: To solve the correct equation, $$x^2 + 3x - 550 = 0$$, you can use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$.

Q: What is the correct solution for $$x$$?

A: The correct solution for $$x$$ is $$x = -25$$.

Q: Can I use other methods to solve the quadratic equation?

A: Yes, you can use other methods to solve the quadratic equation, such as factoring, graphing, or using the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations in the form $$ax^2 + bx + c = 0$$. It is given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$.

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula, you need to identify the values of $$a$$, $$b$$, and $$c$$ in the quadratic equation. Then, you can plug these values into the quadratic formula and simplify to find the solutions for $$x$$.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

Not identifying the correct values of $$a$$, $$b$$, and $$c$$ in the quadratic equation.
Not using the correct formula to solve the quadratic equation.
Not simplifying the expression under the square root.
Not checking for extraneous solutions.

Q: Can I use technology to solve quadratic equations?

A: Yes, you can use technology, such as calculators or computer software, to solve quadratic equations. However, it's still important to understand the underlying mathematics and be able to solve the equation by hand.

Conclusion

In conclusion, solving quadratic equations and finding the smaller number requires a clear understanding of the underlying mathematics. By using the quadratic formula and avoiding common mistakes, you can solve quadratic equations and find the value of the smaller number. Remember to always check your work and be careful when simplifying expressions.

Frequently Asked Questions (FAQs)

What is the product of two numbers, $$x$$ and $$x+3$$?
How do I solve the correct equation to find the value of the smaller number, $$x$$?
What is the correct solution for $$x$$?
Can I use other methods to solve the quadratic equation?
What is the quadratic formula?
How do I use the quadratic formula to solve a quadratic equation?
What are some common mistakes to avoid when solving quadratic equations?
Can I use technology to solve quadratic equations?

Additional Resources

Quadratic Formula: A quadratic equation in the form $$ax^2 + bx + c = 0$$ can be solved using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$.
Solving Quadratic Equations: There are several methods for solving quadratic equations, including factoring, the quadratic formula, and graphing.
Algebra: Algebra is a branch of mathematics that deals with the study of variables and their relationships. It includes topics such as linear equations, quadratic equations, and functions.