When Solving The Equation $\frac{x-2}{6}=\frac{1}{x+3}$, Santos Starts By Cross-multiplying To Get $6=x^2+x-6$.Then, He Subtracts 6 From Both Sides Of The Equation To Result In $x^2+x-12=0$.What Can Santos Do Next To Solve

Introduction

Solving quadratic equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy a quadratic equation. In this article, we will explore the steps involved in solving quadratic equations, using the example of Santos, who is trying to solve the equation $\frac{x-2}{6}=\frac{1}{x+3}$.

Understanding the Problem

Santos starts by cross-multiplying to get $6=x^2+x-6$. This is a quadratic equation in the form of $ax^2+bx+c=0$, where $a=1$, $b=1$, and $c=-12$. The goal is to find the values of $x$ that satisfy this equation.

The Quadratic Formula

One way to solve quadratic equations is by using the quadratic formula, which is given by:

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

In this case, $a=1$, $b=1$, and $c=-12$. Plugging these values into the quadratic formula, we get:

$$x=\frac{-1\pm\sqrt{1^2-4(1)(-12)}}{2(1)}$$

Simplifying the expression under the square root, we get:

$$x=\frac{-1\pm\sqrt{1+48}}{2}$$

$$x=\frac{-1\pm\sqrt{49}}{2}$$

$$x=\frac{-1\pm7}{2}$$

This gives us two possible values for $x$: $x=\frac{-1+7}{2}$ and $x=\frac{-1-7}{2}$.

Solving for $x$

Let's solve for $x$ in both cases.

Case 1: $x=\frac{-1+7}{2}$

$$x=\frac{-1+7}{2}$$

$$x=\frac{6}{2}$$

$$x=3$$

Case 2: $x=\frac{-1-7}{2}$

$$x=\frac{-1-7}{2}$$

$$x=\frac{-8}{2}$$

$$x=-4$$

Conclusion

In this article, we have explored the steps involved in solving quadratic equations using the quadratic formula. We have used the example of Santos, who is trying to solve the equation $\frac{x-2}{6}=\frac{1}{x+3}$. By applying the quadratic formula, we have found two possible values for $x$: $x=3$ and $x=-4$. These values satisfy the original equation, and we can verify this by plugging them back into the equation.

Tips and Tricks

Here are some tips and tricks to keep in mind when solving quadratic equations:

• Check your work: Always check your work by plugging the values back into the original equation.
• Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations.
• Simplify the expression: Simplify the expression under the square root to make it easier to work with.
• Check for extraneous solutions: Check for extraneous solutions by plugging the values back into the original equation.

Common Mistakes

Here are some common mistakes to avoid when solving quadratic equations:

• Not checking your work: Failing to check your work can lead to incorrect solutions.
• Not using the quadratic formula: Failing to use the quadratic formula can make it difficult to solve the equation.
• Not simplifying the expression: Failing to simplify the expression under the square root can make it difficult to work with.
• Not checking for extraneous solutions: Failing to check for extraneous solutions can lead to incorrect solutions.

Real-World Applications

Quadratic equations have many real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model economic systems and make predictions about future trends.
• Computer Science: Quadratic equations are used in computer graphics and game development.

Conclusion

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Frequently Asked Questions

Q: What is a quadratic equation?

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

Q: How do I solve a quadratic equation?

A: There are several ways to solve a quadratic equation, including:

• Factoring: If the quadratic expression can be factored into the product of two binomials, you can set each binomial equal to zero and solve for x.
• Quadratic formula: The quadratic formula is a formula that can be used to solve any quadratic equation. It is given by x = (-b ± √(b^2 - 4ac)) / 2a.
• Graphing: You can also solve a quadratic equation by graphing the related function and finding the x-intercepts.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve any quadratic equation. It is given by x = (-b ± √(b^2 - 4ac)) / 2a.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of a, b, and c from the quadratic equation into the formula. Then, simplify the expression under the square root and solve for x.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a squared variable (x^2), while a linear equation does not.

Q: Can I solve a quadratic equation by hand?

A: Yes, you can solve a quadratic equation by hand using the quadratic formula or factoring. However, for more complex equations, it may be easier to use a calculator or computer software.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have many real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model economic systems and make predictions about future trends.
• Computer Science: Quadratic equations are used in computer graphics and game development.

Q: Can I use a calculator to solve a quadratic equation?

A: Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in quadratic formula function that you can use to solve the equation.

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

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

• Not checking your work: Failing to check your work can lead to incorrect solutions.
• Not using the quadratic formula: Failing to use the quadratic formula can make it difficult to solve the equation.
• Not simplifying the expression: Failing to simplify the expression under the square root can make it difficult to work with.
• Not checking for extraneous solutions: Failing to check for extraneous solutions can lead to incorrect solutions.

Q: Can I use a computer program to solve a quadratic equation?

A: Yes, you can use a computer program to solve a quadratic equation. Many computer programs, such as MATLAB and Python, have built-in functions for solving quadratic equations.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics that have many real-world applications. By understanding how to solve quadratic equations, you can apply this knowledge to a wide range of fields, including physics, engineering, economics, and computer science. Remember to check your work, use the quadratic formula, simplify the expression, and check for extraneous solutions to ensure that you are getting the correct solutions.