Problem 11 - Beginning and Intermediate Algebra

Question

Solve. $$x+\frac{6}{x}=5$$

Step-by-Step Solution

Answer

The solution to the equation \(x + \frac{6}{x} = 5\) is \(x = 2\) or \(x = 3\).

Step 1: Rewrite the equation

Rewrite the given equation: \(x + \frac{6}{x} = 5\)

Step 2: Clear the fraction

Multiply both sides of the equation by \(x\): \(x(x + \frac{6}{x}) = 5x\)

Step 3: Distribute and simplify

Distribute and simplify the equation: \(x^2 + 6 = 5x\)

Step 4: Rearrange the equation into a quadratic form

Subtract \(5x\) from both sides of the equation to get a quadratic equation: \(x^2 - 5x + 6 = 0\)

Step 5: Factor the quadratic equation

Factor the equation: \((x - 2)(x - 3) = 0\)

Step 6: Solve for x

Apply the zero-product property to find the values of \(x\): \(x - 2 = 0\) or \(x - 3 = 0\) \(x = 2\) or \(x = 3\) The solution to the equation \(x + \frac{6}{x} = 5\) is \(x = 2\) or \(x = 3\).

Key Concepts

Solving Equations

When it comes to solving equations, our goal is to find the value of the variable that makes the equation true. Let’s take the equation from the exercise, where we have to solve \(x + \frac{6}{x} = 5\). To do this, we work through a series of steps:

• Clear fractions: Start by eliminating the fraction by multiplying all terms by \(x\). This simplifies the equation, making it easier to handle.
• Simplify: Next, distribute the \(x\) and rearrange terms to isolate the variable and turn the equation quadratic.
• Solve: Lastly, apply mathematical strategies such as factoring or using the quadratic formula to find your solutions.

By following these steps carefully, solving equations becomes a manageable task. Remember, practice makes perfect, so keep trying with different problems!

Factoring Quadratics

Factoring quadratics is a critical skill when solving quadratic equations. Quadratic equations are generally in the form \(ax^2 + bx + c = 0\). Our job is to express this as a product of two binomials if possible. For example, in our exercise, we rearranged our equation to obtain the quadratic form: \[ x^2 - 5x + 6 = 0 \] In this case, we look for two numbers that multiply to the constant term, 6, and add to the linear coefficient, -5. These numbers are -2 and -3, allowing us to factor the equation as: \[(x - 2)(x - 3) = 0\] Some key tips for factoring quadratics include:

• First, check for common factors before moving to more complex methods.
• Remember that not all quadratics are easily factorable, so sometimes using the quadratic formula or completing the square is necessary.

Factoring quadratics requires some practice, but understanding the principles will help tremendously.

Zero-Product Property

The zero-product property is a handy tool when solving quadratic equations. It states that if the product of two factors is zero, then at least one of the factors must be zero. This property is hugely beneficial when you have factored a quadratic equation. In our example from the step-by-step solution: \[(x - 2)(x - 3) = 0\] We use the zero-product property to solve for \(x\). This implies:

• If \((x - 2) = 0\), then \(x = 2\).
• If \((x - 3) = 0\), then \(x = 3\).

Therefore, the solutions to the equation are \(x = 2\) and \(x = 3\). Using this property brings us closer to solving the equation efficiently. Remember, the zero-product property only works if the multiplication of terms gives zero, so always ensure your equation is properly factored down.