Solve And Check:$\frac{1}{x+3}=\frac{x+10}{x-2}$From Least To Greatest, The Solutions Are $x = \square$ And $x = \square$.

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Introduction

In this article, we will delve into solving and checking the given equation 1x+3=x+10x−2\frac{1}{x+3}=\frac{x+10}{x-2}. We will follow a step-by-step approach to find the solutions and then arrange them in order from least to greatest.

Step 1: Cross-Multiplication

To solve the equation, we will start by cross-multiplying both sides of the equation. This will help us eliminate the fractions and make it easier to work with.

1x+3=x+10x−2\frac{1}{x+3}=\frac{x+10}{x-2}

Cross-multiplying:

1(x−2)=(x+10)(x+3)1(x-2) = (x+10)(x+3)

Step 2: Expanding and Simplifying

Now, let's expand and simplify the equation.

1(x−2)=(x+10)(x+3)1(x-2) = (x+10)(x+3)

Expanding:

x−2=x2+13x+30x - 2 = x^2 + 13x + 30

Simplifying:

0=x2+13x+30−(x−2)0 = x^2 + 13x + 30 - (x - 2)

0=x2+13x+30−x+20 = x^2 + 13x + 30 - x + 2

0=x2+12x+320 = x^2 + 12x + 32

Step 3: Factoring the Quadratic Equation

Next, we will try to factor the quadratic equation.

0=x2+12x+320 = x^2 + 12x + 32

Factoring:

0=(x+8)(x+4)0 = (x + 8)(x + 4)

Step 4: Finding the Solutions

Now, we can find the solutions by setting each factor equal to zero.

0=(x+8)(x+4)0 = (x + 8)(x + 4)

Setting each factor equal to zero:

x+8=0x + 8 = 0 or x+4=0x + 4 = 0

Solving for x:

x=−8x = -8 or x=−4x = -4

Step 5: Checking the Solutions

To check the solutions, we will substitute each value back into the original equation.

Substituting x=−8x = -8:

1(−8)+3=(−8)+10(−8)−2\frac{1}{(-8)+3}=\frac{(-8)+10}{(-8)-2}

1−5=2−10\frac{1}{-5}=\frac{2}{-10}

−15=−15- \frac{1}{5} = - \frac{1}{5}

Substituting x=−4x = -4:

1(−4)+3=(−4)+10(−4)−2\frac{1}{(-4)+3}=\frac{(-4)+10}{(-4)-2}

1−1=6−6\frac{1}{-1}=\frac{6}{-6}

−1=−1-1 = -1

Conclusion

Introduction

In our previous article, we solved and checked the equation 1x+3=x+10x−2\frac{1}{x+3}=\frac{x+10}{x-2}. We found the solutions by cross-multiplying, expanding and simplifying, factoring the quadratic equation, and finding the solutions. We then checked the solutions by substituting each value back into the original equation. In this article, we will answer some frequently asked questions related to the equation and its solutions.

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to cross-multiply both sides of the equation. This will help us eliminate the fractions and make it easier to work with.

Q: Why do we need to expand and simplify the equation?

A: We need to expand and simplify the equation to make it easier to work with and to identify the quadratic equation.

Q: How do we factor the quadratic equation?

A: We factor the quadratic equation by finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: What are the solutions to the equation?

A: The solutions to the equation are x=−8x = -8 and x=−4x = -4.

Q: How do we check the solutions?

A: We check the solutions by substituting each value back into the original equation.

Q: What if the equation has no real solutions?

A: If the equation has no real solutions, it means that the quadratic equation has no real roots. In this case, we would need to use complex numbers to find the solutions.

Q: Can we use other methods to solve the equation?

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

Q: What is the importance of checking the solutions?

A: Checking the solutions is important to ensure that the solutions are correct and to avoid making mistakes.

Q: Can we use technology to solve the equation?

A: Yes, we can use technology, such as calculators or computer software, to solve the equation and find the solutions.

Conclusion

In this article, we answered some frequently asked questions related to the equation and its solutions. We hope that this article has been helpful in clarifying any doubts or questions you may have had.

Additional Resources

Frequently Asked Questions

  • 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.
  • Q: How do I know if an equation is quadratic or linear? A: You can determine if an equation is quadratic or linear by looking at the degree of the polynomial.
  • Q: Can I use the quadratic formula to solve any quadratic equation? A: Yes, you can use the quadratic formula to solve any quadratic equation.

Glossary

  • Quadratic equation: A polynomial equation of degree two.
  • Linear equation: A polynomial equation of degree one.
  • Complex numbers: Numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.
  • Quadratic formula: A formula used to solve quadratic equations, which is x = (-b ± √(b^2 - 4ac)) / 2a.