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Chapter 1: Problem 34

Solve equation by the square root property. $$ (2 x+8)^{2}=27 $$

Short Answer

Expert verified
The solutions are \(x = \frac{\sqrt{27} - 8}{2}\) and \(x = \frac{-\sqrt{27} - 8}{2}\)

Step by step solution

01

Isolate the quadratic term

Initially, the quadratic term, \((2x+8)^2\), is isolated. This isn't necessary for this problem, as the equation is already set equal to a constant, 27.
02

Apply the square root property

Next, take the square root of both sides of the equation. This leads to: \[ 2x+8 = 卤 \sqrt{27} \]Remember to include both the positive and negative square roots as the square root property gives us two solutions.
03

Solve for the variable

Finally, solve for the variable x. This can be done by initially subtracting 8 from both sides, leading to \(2x = 卤 \sqrt{27} - 8\). The last step is to divide by two, arriving at: \[ x = \frac{卤 \sqrt{27} - 8}{2} \]

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Key Concepts

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

Quadratic Equation
A quadratic equation is a type of polynomial equation that involves terms up to the second power. You can recognize it by its general formula:
  • ax虏 + bx + c = 0
Here, 'a', 'b', and 'c' are constants, with 'a' not equal to zero. The quadratic equation in our exercise is slightly different. It's given in the form of
  • (2x + 8)虏 = 27
This equation is already presented with the squared term isolated, which makes it a perfect candidate for using the square root property directly. Understanding these structures helps simplify the process of solving quadratic equations and clarifying what solutions to expect.
Solving Equations
Solving an equation means finding the value or values of the variable that make the equation true. For quadratic equations, this can involve a few different methods, such as factoring, completing the square, or using the quadratic formula. In some special cases, like our example, we utilize the square root property.
The process typically involves:
  • Isolating the variable terms on one side of the equation.
  • Eliminating constants and coefficients.
  • Utilizing appropriate methods to find the variable values.
Solving equations requires careful manipulation to maintain equality on both sides, ensuring that the operations result in a mathematically valid solution.
Square Roots
Square roots are mathematical operations that determine what number, when multiplied by itself, produces the original number. The square root of a number 'n' is expressed as
  • 鈭歯
When dealing with square roots in equations, particularly quadratic ones, it鈥檚 crucial to consider both the positive and negative roots.
This is because:
  • Both square a positive and a negative number result in the same product: (鈭歯)虏 = n and (-鈭歯)虏 = n.
In our exercise,
  • 鈭�27
results in 卤鈭�27, introducing two possible solutions to explore further. Understanding how square roots behave with numbers helps solve equations more effectively.
Variable Isolation
Variable isolation refers to the process of manipulating an equation to get the variable of interest by itself on one side of the equation. This step is essential to find the actual value(s) of the variable.
For instance, in our equation
  • (2x + 8) = 卤鈭�27
we isolate 'x' by performing operations to eliminate other terms:
  • Subtract 8 from both sides: 2x = 卤鈭�27 - 8
  • Divide each side by 2: x = (卤鈭�27 - 8) / 2
Employing operations like addition, subtraction, multiplication, and division enables us to isolate variables systematically. This helps us solve for them more directly and clearly.

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Most popular questions from this chapter

In Exercises 122鈥�133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. You are choosing between two texting plans. Plan A has a monthly fee of 15 dollar with a charge of 0.08 dollar per text. Plan \(B\) has a monthly fee of 3 dollar with a charge of 0.12 dollar per text. How many text messages in a month make plan A the better deal?

In a round-robin chess tournament, each player is paired with every other player once. The formula $$ N=\frac{x^{2}-x}{2} $$ models the number of chess games, \(N,\) that must be played in a round-robin tournament with \(x\) chess players. Use this formula to solve. In a round-robin chess tournament, 36 games were played. How many players were entered in the tournament?

Explaining the Concepts. Describe the solution set of \(|x|>-4\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I obtained \(-17\) for the discriminant, so there are two imaginary irrational solutions.

What is a quadratic equation?
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