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Chapter 1: Problem 98

Solve equation by the method of your choice. $$ 3 x^{2}-27=0 $$

Short Answer

Expert verified
The solutions to the equation \(3x^{2} - 27 = 0\) are \(x = -3\) and \(x = 3\).

Step by step solution

01

Rearrange the equation

First, let's rearrange the equation in order to make it easier to solve. We can start by adding 27 to both sides to get the equation: \(3x^{2} = 27\).
02

Solve for \(x^{2}\)

Next, we can divide both sides by 3 to isolate \(x^{2}\). This gives us the equation: \(x^{2} = 9\).
03

Solve for \(x\)

To find \(x\), we take the square root of both sides. This gives us two solutions since the square root of a number can be both positive and negative. Therefore, the solutions to the equation are \(x = -3\) and \(x = 3\).

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Key Concepts

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

Square Root Method
When solving quadratic equations, a variety of methods can be employed, and one of the simplest approaches is the Square Root Method. This method is especially useful when the quadratic equation is already in the form of \(ax^2 = c\). Here鈥檚 how it works:
  • First, isolate the \(x^2\) term by moving all other terms to the other side of the equation. This step makes it easier to take the square root later.
  • Once the equation is in the form \(x^2 = c\), the next step is to apply the square root to both sides of the equation.
  • Remember that whenever we take the square root of an equation, there are two potential solutions: one positive and one negative. This is due to the properties of squares in algebra. For example, both \(3^2\) and \((-3)^2\) equal 9.
This method is straightforward and effective, though it only works directly when the quadratic equation is suited to it, typically when there is no linear \(x\) term present.
Solving Equations
Solving equations is a central aspect of algebra and involves finding the value or values of variables that make the equation true. Here鈥檚 a quick guide to understanding this process:
  • Identify your equation type: Linear, quadratic, or others. In this instance, we focus on quadratic equations.
  • Choose a method based on the equation type and form. For a simple quadratic equation like \(3x^2 = 27\), isolating \(x^2\) and applying the square root method is effective.
  • Perform algebraic operations step-by-step, such as addition, subtraction, multiplication, or division, to rearrange the equation.
  • Check your solutions by substituting them back into the original equation to ensure they satisfy the equation鈥檚 conditions.
Applying these steps systematically allows for solving equations efficiently and confidently.
Algebraic Manipulation
Algebraic manipulation is the process of rearranging, simplifying, or rewriting equations to make them more manageable or solve them. Here鈥檚 how it works:
  • Start by identifying terms and operators within the equation. Here, in \(3x^2 - 27 = 0\), the goal is to simplify or isolate the terms.
  • Use arithmetic inversion, like adding or subtracting values across the equation, to move terms and simplify the equation structurally. For example, adding 27 to both sides to get \(3x^2 = 27\).
  • Apply multiplication or division to further simplify. Dividing both sides by 3 simplifies the equation to \(x^2 = 9\).
  • Throughout the process, maintain balance by performing the same operations on both sides of the equation, ensuring equality is preserved.
This approach allows for solving more complex problems by breaking them down into simpler, manageable steps, facilitating easier problem-solving in algebra.

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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. An elevator at a construction site has a maximum capacity of 2800 pounds. If the elevator operator weighs 265 pounds and each cement bag weighs 65 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

In Exercises 122鈥�133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells personalized stationery. The weekly fixed cost is 3000 dollar and it costs 3.00 dollar to produce each package of stationery. The selling price is $5.50 per package. How many packages of stationery must be produced and sold each week for the company to generate a profit?

In Exercises 122鈥�133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells blank audio cassette tapes. The weekly fixed cost is 10.000 dollar and it costs 0.40 dollar to produce each tape. The selling price is 2.00 dollar per tape. How many tapes must be produced and sold each week for the company to generate a profit?

What is a quadratic equation?

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, 21 games were played. How many players were entered in the tournament?
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