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Chapter 7: Problem 17

Solve each rational equation. $$x+\frac{6}{x}=-7$$

Short Answer

Expert verified
The solutions to the equation are \(x = -1\) and \(x = -6\)

Step by step solution

01

Eliminate the fractions

This would involve getting rid of the fraction by multiplying each term by \(x\). This gets rid of x in the denominator. Doing this yields: \(x^2 + 6 = -7x \)
02

Rearrange the equation

Rearrange the equation by bringing the terms to one side of the equation to set it equal to zero. This gives us a quadratic equation: \(x^2 +7x +6 = 0 \)
03

Factor the quadratic equation

This can be factored into \((x +1)(x + 6) = 0\)
04

Find the roots of the equation

Finally, set each factor equal to zero and solve for x to obtain the roots of the equation: \(x= -1, -6\)

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Key Concepts

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

Understanding Quadratic Equations
Quadratic equations are a type of polynomial equation where the highest power of the variable is squared, which is why they're called "quadratic." The general form of a quadratic equation is:\[ ax^2 + bx + c = 0 \]where:
  • \(a, b,\) and \(c\) are constants
  • \(x\) is the variable
  • \(a\) cannot be zero
Quadratic equations model many real-world situations, such as predicting projectile motion and understanding area problems. In our exercise, we rearranged the rational equation into a quadratic form, \(x^2 + 7x + 6 = 0\), to make it easier to solve.
Factoring Quadratic Equations
Factoring is a method of breaking down expressions into simpler 'factors' which are multiplied together to give the original expression. For quadratic equations, factoring can help us find the solutions or 鈥榬oots鈥�. The process involves:
  • Finding two numbers that multiply to give the constant term \(c\) and add up to give the middle term \(b\).
  • Rewriting the quadratic as a product of two binomials.
In our given equation, \(x^2 + 7x + 6 = 0\), we look for numbers that multiply to 6 (the constant) and add to 7 (the middle term). Here, the numbers 1 and 6 fit perfectly, so we factor the equation as \((x + 1)(x + 6) = 0\). This makes it simpler to find the solutions.
Finding the Roots of Equations
The "roots" of an equation are the values of the variable that satisfy the equation. For quadratic equations, these roots can be found by factoring as we did, completing the square, or using the quadratic formula. When an equation is factored like \((x + 1)(x + 6) = 0\), finding its roots is straightforward.Set each factor equal to zero:
  • \(x + 1 = 0\) leads to \(x = -1\)
  • \(x + 6 = 0\) leads to \(x = -6\)
Thus, the roots or solutions of the quadratic equation \(x^2 + 7x + 6 = 0\) are \(x = -1\) and \(x = -6\). These roots represent the x-values where the equation holds true. Understanding this fundamental concept connects algebraic manipulation to the solution of practical problems.

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Most popular questions from this chapter

Add or subtract as indicated. Simplify the result, if possible. $$\frac{2}{x}+9$$

Explain how to add rational expressions when denominators are the same. Give an example with your explanation.

It normally takes 2 hours to fill a swimming pool. The pool has developed a slow leak. If the pool were full, it would take 10 hours for all the water to leak out. If the pool is empty, how long will it take to fill it?

Will help you prepare for the material covered in the next section. a. If \(y=k x\) find the value of \(k\) using \(x=2\) and \(y=64\) b. Substitute the value for \(k\) into \(y=k x\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=5\)

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The difference between two rational expressions with the same denominator can always be simplified.
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