Question

Determine the solution to the equation \(\frac{2 x+1}{x-1}=\frac{2}{x+2}-\frac{3}{2}\) using two different methods.

Short answer

No real solutions; complex solutions: \(x = \frac{-7 \pm i \sqrt{7}}{14}\).

Step-by-step solution

- Simplify the Equation

First, rewrite the equation in a more manageable form. Start from the given equation: \[\frac{2x + 1}{x - 1} = \frac{2}{x + 2} - \frac{3}{2}\]Combine the terms on the right-hand side: \[\frac{2x + 1}{x - 1} = \frac{4 - 3(x + 2)}{2(x + 2)} = \frac{4 - 3x - 6}{2(x + 2)} = \frac{-3x - 2}{2(x + 2)}\]

- Cross-Multiply to Eliminate Fractions

To eliminate the fractions, cross-multiply: \[2(x + 2)(2x + 1) = (x - 1)(-3x - 2)\]This gives us: \[4x^2 + 4x + 4x + 4 = -3x^2 - 2x + 3x + 2\]

- Simplify and Combine Like Terms

Combine like terms on both sides of the equation: \[4x^2 + 8x + 4 = -3x^2 + x + 2\]Move everything to one side of the equation to set it to zero: \[4x^2 + 8x + 4 + 3x^2 - x - 2 = 0\]Combine the terms: \[7x^2 + 7x + 2 = 0\]

- Solve the Quadratic Equation using the Quadratic Formula

The quadratic equation is in the form \(ax^2 + bx + c = 0\), where \(a = 7\), \(b = 7\) and \(c = 2\). Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve for x.Substitute the values into the formula: \[x = \frac{-7 \pm \sqrt{7^2 - 4(7)(2)}}{2(7)} = \frac{-7 \pm \sqrt{49 - 56}}{14} = \frac{-7 \pm \sqrt{-7}}{14}\]Since \(\sqrt{-7}\) has no real value, the equation has no real solutions.

- Verify if Complex Solutions are Valid

The quadratic formula solution indicates the presence of complex solutions due to the negative discriminant: \[x = \frac{-7 \pm i \sqrt{7}}{14}\] If a deeper exploration of complex solutions is required, consider different contexts of use.

Key concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find solutions for quadratic equations. Quadratic equations are typically in the form \(ax^2 + bx + c = 0\). The formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a\), \(b\), and \(c\) are coefficients from the quadratic equation.

Let's break it down:

• \(b^2 - 4ac\) is called the 'discriminant'. It determines the nature of the roots.
• The \(\pm\) symbol means there are typically two solutions.

In our exercise, we used the quadratic formula and found that \( b^2 - 4ac\) was negative, indicating no real solutions.

Instead, we got complex solutions.

Complex Numbers

Complex numbers are used when equations have roots that aren't real. They are written in the form \(a + bi\) where:

• \(a\) is the real part.
• \(bi\) is the imaginary part where \(i\) is the square root of -1.

In the given exercise, when solving using the quadratic formula, \( \sqrt{-7} \) was obtained, indicating the use of complex numbers.

This resulted in the solutions: \[ x = \frac{-7 \pm i \sqrt{7}}{14} \]
This is an example of how complex numbers are used to represent solutions for equations not solvable in the real number set.

Cross-Multiplication

Cross-multiplication is a method used to remove fractions from equations. This technique is particularly useful when dealing with rational equations.

Here’s how it works:

• Given an equation in the form \frac{A}{B} = \frac{C}{D}\, we cross-multiply to get: \A \cdot D = B \cdot C\.
• This process helps to simplify the equation by removing the denominators.

In the exercise, cross-multiplication was applied to:
\frac{2x + 1}{x - 1} = \frac{-3x - 2}{2(x + 2)}\.

Resulting in: \[2(x + 2)(2x + 1) = (x - 1)(-3x - 2)\ \]

Combining Like Terms

Combining like terms simplifies equations by grouping similar expressions. This helps in making the equations more manageable.

Here's a quick breakdown:

• Identify terms with the same variable and exponent.
• Add or subtract the coefficients of those terms.

In the given problem, after cross-multiplying, we had: \4x^2 + 8x + 4 = -3x^2 + x + 2\.

Combining like terms meant: \4x^2 + 3x^2 + 8x - x + 4 - 2 = 0\,
giving us a simplified equation: \[ ^7x^2 + 7x + 2 = 0 \]
This made it easier to apply the quadratic formula for finding solutions.