Solve The Equation $2x - 3 = \frac{112}{3x + 1}$.If You Have More Than One Solution, Separate Them With Commas As Reduced Fractions Or Integers. X = X = X = □ \square □

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Introduction

In this article, we will be solving a complex equation involving fractions and variables. The equation is given as $2x - 3 = \frac{112}{3x + 1}$, and we need to find the value of x that satisfies this equation. We will use algebraic techniques to solve this equation and find the possible values of x.

Step 1: Multiply Both Sides by the Denominator

To eliminate the fraction, we can multiply both sides of the equation by the denominator, which is 3x+13x + 1. This will give us:

2x−3=1123x+12x - 3 = \frac{112}{3x + 1}

⇒(2x−3)(3x+1)=112\Rightarrow (2x - 3)(3x + 1) = 112

Step 2: Expand the Left Side of the Equation

Now, we can expand the left side of the equation using the distributive property:

(2x−3)(3x+1)=6x2−9x+2x−3(2x - 3)(3x + 1) = 6x^2 - 9x + 2x - 3

⇒6x2−7x−3=112\Rightarrow 6x^2 - 7x - 3 = 112

Step 3: Rearrange the Equation

Next, we can rearrange the equation to get a quadratic equation in the form of ax2+bx+c=0ax^2 + bx + c = 0:

6x2−7x−3−112=06x^2 - 7x - 3 - 112 = 0

⇒6x2−7x−115=0\Rightarrow 6x^2 - 7x - 115 = 0

Step 4: Solve the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, a=6a = 6, b=−7b = -7, and c=−115c = -115. Plugging these values into the formula, we get:

x=−(−7)±(−7)2−4(6)(−115)2(6)x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(6)(-115)}}{2(6)}

⇒x=7±49+276012\Rightarrow x = \frac{7 \pm \sqrt{49 + 2760}}{12}

⇒x=7±280912\Rightarrow x = \frac{7 \pm \sqrt{2809}}{12}

Step 5: Simplify the Solutions

Now, we can simplify the solutions by evaluating the square root:

2809=53\sqrt{2809} = 53

So, the solutions are:

x=7±5312x = \frac{7 \pm 53}{12}

Step 6: Find the Possible Values of x

Now, we can find the possible values of x by evaluating the two expressions:

x=7+5312=6012=5x = \frac{7 + 53}{12} = \frac{60}{12} = 5

x=7−5312=−4612=−236x = \frac{7 - 53}{12} = \frac{-46}{12} = -\frac{23}{6}

Conclusion

In this article, we solved the equation $2x - 3 = \frac{112}{3x + 1}$ and found the possible values of x. The solutions are x=5x = 5 and x=−236x = -\frac{23}{6}. We used algebraic techniques to solve this equation and found the values of x that satisfy the equation.

Final Answer

The final answer is: 5,−236\boxed{5, -\frac{23}{6}}

Introduction

In our previous article, we solved the equation $2x - 3 = \frac{112}{3x + 1}$ and found the possible values of x. In this article, we will answer some frequently asked questions related to solving this equation.

Q: What is the first step in solving the equation 2x - 3 = 112 / (3x + 1)?

A: The first step in solving the equation is to multiply both sides of the equation by the denominator, which is 3x+13x + 1. This will eliminate the fraction and make it easier to solve the equation.

Q: Why do we need to multiply both sides of the equation by the denominator?

A: We need to multiply both sides of the equation by the denominator to eliminate the fraction. This is because the fraction is not in its simplest form, and multiplying both sides by the denominator will allow us to simplify the equation and solve for x.

Q: What is the quadratic formula, and how is it used to solve the equation?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In the case of the equation $6x^2 - 7x - 115 = 0$, we can use the quadratic formula to find the possible values of x.

Q: What are the possible values of x that satisfy the equation?

A: The possible values of x that satisfy the equation are x=5x = 5 and x=−236x = -\frac{23}{6}. These values can be found by evaluating the two expressions:

x=7+5312=6012=5x = \frac{7 + 53}{12} = \frac{60}{12} = 5

x=7−5312=−4612=−236x = \frac{7 - 53}{12} = \frac{-46}{12} = -\frac{23}{6}

Q: How do we know which value of x is the correct solution?

A: To determine which value of x is the correct solution, we need to plug each value back into the original equation and check if it is true. If the value satisfies the equation, then it is the correct solution.

Q: What if the equation has no real solutions?

A: If the equation has no real solutions, then the quadratic formula will give us complex solutions. In this case, we need to check if the complex solutions are valid or not.

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

A: Yes, we can use other methods to solve the equation, such as factoring or using the quadratic formula. However, the quadratic formula is a powerful tool that can be used to solve quadratic equations, and it is often the most efficient method.

Conclusion

In this article, we answered some frequently asked questions related to solving the equation $2x - 3 = \frac{112}{3x + 1}$. We discussed the first step in solving the equation, the quadratic formula, and the possible values of x that satisfy the equation. We also discussed how to determine which value of x is the correct solution and what to do if the equation has no real solutions.

Final Answer

The final answer is: 5,−236\boxed{5, -\frac{23}{6}}