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Chapter 11: Problem 2

Solve. \(z+11=-\frac{24}{z}\)

Short Answer

Expert verified
The short answer based on the step-by-step solution is: To solve the equation \(z + 11 = -\frac{24}{z}\), first eliminate the fraction by multiplying both sides by z. Next, simplify the equation to get \(z^2 + 11z + 24 = 0\). Factor the quadratic equation as \((z + 3)(z + 8) = 0\) and apply the zero-product property to find the possible values of z: \(z = -3\) and \(z = -8\).

Step by step solution

01

Write the given equation

Start by writing down the equation as given: \(z + 11 = -\frac{24}{z}\) #Step 2: Eliminate the fraction#
02

Eliminate the fraction

To eliminate the fraction, multiply both sides of the equation by z: \(z(z + 11) = -\frac{24}{z} * z\) #Step 3: Simplify the equation#
03

Simplify the equation

Now, simplify both sides of the equation: \(z^2 + 11z = -24\) #Step 4: Move all terms to one side of the equation to create a quadratic equation#
04

Combine terms to form a quadratic equation

Move the -24 to the left side of the equation to get the quadratic equation: \(z^2 + 11z + 24 = 0\) #Step 5: Factor the quadratic equation#
05

Factor the quadratic equation

Try to factor the quadratic equation to find the factors of z: \((z + 3)(z + 8) = 0\) #Step 6: Solve for z#
06

Solve for the variable z

Now, apply the zero-product property to solve for z: 1. \(z + 3 = 0\) which implies that \(z = -3\) 2. \(z + 8 = 0\) which implies that \(z = -8\) So, we have two possible values for z: \(z = -3\) and \(z =-8\).

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Key Concepts

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

Factoring Quadratic Equations
Factoring quadratic equations is a crucial skill in algebra. When you have a quadratic equation like \(z^2 + 11z + 24 = 0\), factoring is a method that allows you to express the quadratic expression as a product of two binomials.
Here are some tips to aid in factoring:
  • Look for two numbers that multiply to give the constant term (in this case, 24) and add up to the coefficient of the middle term (11).
  • For this exercise, the numbers 3 and 8 fit the criteria, as 3 \(\times\) 8 = 24 and 3 + 8 = 11.
  • Thus, the quadratic equation can be factored as \((z + 3)(z + 8) = 0\).
Factoring can be tricky if factors aren't immediately obvious, but with practice, it becomes easier and can be a quick way to solve quadratic equations.
Zero-Product Property
The zero-product property is a fundamental principle used in solving factored quadratic equations. It states that if the product of two numbers is zero, then at least one of the numbers must be zero. This principle is crucial when you factor a quadratic equation into the form \((z+a)(z+b)=0\).
Applying this property implies:
  • Set each binomial factor in the equation equal to zero.
  • For example, from \((z + 3)(z + 8) = 0\), we get two separate equations: \(z + 3 = 0\) and \(z + 8 = 0\).
  • Solving these gives the possible solutions, \(z = -3\) and \(z = -8\), which are the roots of the original quadratic equation.
Using the zero-product property is a straightforward step that simplifies finding solutions to quadratic equations after factoring.
Eliminating Fractions
Eliminating fractions is often the first step to simplify an equation before solving it, especially when variables are in denominators. In this exercise, you have an equation like \(z + 11 = -\frac{24}{z}\). Here鈥檚 how you can clear the fraction:
  • Multiply every term in the equation by the denominator of the fraction (which is \(z\) here) to eliminate the fraction.
  • This changes the equation to \(z(z + 11) = -24\), where the fraction is now eliminated.
  • Eliminating fractions helps you work with simpler expressions, which makes the solving process easier and clearer.
This essential step transforms complex equations into a more manageable form, paving the way for further simplification.
Simplifying Equations
After eliminating fractions, simplifying the equation is key to solving it efficiently. Once fractions are removed, you aim to rewrite the equation in a form that is most conducive to factoring.
In the given exercise, you initially have:
  • The expression transformed to \(z^2 + 11z = -24\) after eliminating fractions.
  • Moving -24 to the left side, the equation becomes \(z^2 + 11z + 24 = 0\).
  • This restructuring creates a clear quadratic equation ready for factoring.
By simplifying expressions and organizing them into a standard quadratic form \(ax^2 + bx + c = 0\), you make factoring more accessible and solving the equation more straightforward. Each step of simplification brings you closer to finding the solution.

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