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Chapter 3: Problem 11

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

Short Answer

Expert verified
x = 2 or x = 3

Step by step solution

01

Multiply by x

Multiply both sides of the equation by x to eliminate the fraction:\[ x \times x + x \times \frac{6}{x} = 5 \times x \]This simplifies to:\[ x^2 + 6 = 5x \]
02

Rearrange the Equation

Rearrange the equation to set it to zero:\[ x^2 - 5x + 6 = 0 \]
03

Factor the Quadratic Equation

Factor the quadratic equation:\[ x^2 - 5x + 6 = (x - 2)(x - 3) = 0 \]
04

Solve for x

Set each factor equal to zero and solve for x:\[ x - 2 = 0 \Rightarrow x = 2 \]\[ x - 3 = 0 \Rightarrow x = 3 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring
Factoring is an essential technique used in solving quadratic equations. When we factor an equation, we rewrite it as a product of its linear factors. This simplifies finding the solutions because it allows us to solve multiple simpler equations instead of one complex one.
In our example, the original quadratic equation was \(x^2 - 5x + 6 = 0\). Factoring this, we find two binomials that multiply to give the quadratic expression. These two factors are \(x - 2\) and \(x - 3\).
So, \(x^2 - 5x + 6 = (x - 2)(x - 3)\). This transformation makes it easier to solve the equation. We'll see that setting each factor to zero will yield simple, linear solutions.
Quadratic Equation
A quadratic equation is a second-degree polynomial equation in the form \(ax^2 + bx + c = 0\). It typically has two solutions.
In our problem, after the initial manipulation, we obtained the quadratic equation \(x^2 - 5x + 6 = 0\). The term \(x^2\) denotes it as quadratic since its highest power of x is 2.
Quadratic equations can be solved in several ways: factoring, using the quadratic formula, or completing the square. Factoring is often the easiest approach when the quadratic trinomial factors neatly into binomials, as it did in this exercise.
Equation Rearrangement
Rearranging an equation means moving terms from one side to the other to set it up for solving. This usually involves combining like terms and setting the equation to zero.
Initially, we had \(x + \frac{6}{x} = 5\). We multiplied everything by x to clear the fraction, which gave us \(x^2 + 6 = 5x\). To turn it into a standard quadratic equation form, we need all terms on one side. So, we rearranged it to \(x^2 - 5x + 6 = 0\).
By rewriting the equation this way, we can proceed with factoring or other methods to find the solutions.

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