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Chapter 5: Problem 45

Solve each equation. $$2 z(z+6)=2 z^{2}+12 z-8$$

Short Answer

Expert verified
The equation has no solutions.

Step by step solution

01

Expand Left Side

First, expand the equation on the left-hand side: \[2z(z + 6) = 2z^2 + 12z\]
02

Simplify Equation

Now, equate the expanded left side to the right side of the equation: \[2z^2 + 12z = 2z^2 + 12z - 8\] Simplify both sides of the equation by subtracting \(2z^2 + 12z\) from both sides.
03

Isolate Constant

After simplifying, you get: \[0 = -8\]This indicates that the equation does not hold for any value of \(z\).
04

Conclusion

Since the simplified equation \(0 = -8\) is a contradiction, the original equation has no solutions.

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Key Concepts

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

Equation Simplification
Simplifying equations is like cleaning up a messy room. We want to make everything as neat and tidy as possible. In the exercise, we began with the equation \(2z(z+6) = 2z^2 + 12z - 8\). The primary goal of simplification is to reduce the equation so that it becomes easier to work with.

Here's a simple guide to follow when simplifying equations:
  • First, perform operations such as addition, subtraction, multiplication, or division across the equation to consolidate similar terms.
  • Always aim to have all terms involving the variable on one side of the equation.
  • Bring all constants to the other side.


In our exercise, we simplified by equating the left-hand side to the right side. We did this by removing terms common on both sides. Once this is done, we have a clearer view of what's left. Any contradictions found here, like in this exercise, indicate that no solutions exist.
Expanding Expressions
Expanding expressions involves multiplying out expressions into a simpler, clearer form. For instance, in the problem \(2z(z + 6)\), you need to distribute \(2z\) across both terms inside the parentheses.

Follow these steps for expansion:
  • Multiply each term inside the parentheses by the term outside the parentheses.
  • Combine all the resulting terms.
  • Ensure the correct signs (positive/negative) are maintained for each term.


In our exercise, here鈥檚 how it pans out:
  • Multiply \(2z\) by \(z\) to get \(2z^2\).
  • Then, multiply \(2z\) by \(6\) to get \(12z\).

After expanding, the left side of the equation becomes \(2z^2 + 12z\). This process helps in aligning terms on both sides of an equation, which is crucial for further simplification.
No Solution Equations
Not all equations have a neat solution that fits nicely on paper. Sometimes, after all the simplification and calculations, you might end up with a troubling contradiction, such as \(0 = -8\), which we found in this exercise.

When this happens, it indicates that the equation has no solution. Here鈥檚 why it occurs:
  • The equation might represent two expressions that can never be equal because they're essentially parallel like lines that never intersect.
  • The simplification process might reveal that there's no consistent value that satisfies the equation.

It's essential to understand that an equation without a solution is perfectly valid. It simply reflects a scenario where no possible value of a variable will satisfy the conditions laid out by the equation. Once you reach a contradiction, it's the final indication that this equation is unsolvable.

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Most popular questions from this chapter

If \(P(x)=3 x+3, Q(x)=4 x^{2}-6 x+3,\) and \(R(x)=5 x^{2}-7\) find the following. \(2[Q(x)]+7[R(x)]\)

Without calculating, determine which number is larger. $$ 5^{10} \text { or } 5^{9} $$

Factor. Assume that variables used as exponents represent positive integers. $$ x^{2 n}-3 x^{n}-18 $$

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Factor each polynomial completely. See Examples 1 through 12. $$ 12 x^{3}+x^{2}-x $$
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