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Chapter 11: Problem 18

Solve. $$ 7 x^{2}=21 $$

Short Answer

Expert verified
x = 卤鈭�3

Step by step solution

01

Isolate the variable term

First, isolate the term with the variable by dividing both sides of the equation by 7.
02

Simplify both sides

After dividing by 7, simplify the equation to get: \( x^2 = 3 \).
03

Solve for x

Take the square root of both sides to solve for x. Remember to include both the positive and negative roots: \( x = \pm \sqrt{3} \).

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Key Concepts

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

isolating variables
In algebra, isolating variables is a crucial step in solving equations. The main goal is to get the variable you are solving for by itself on one side of the equation.

In the exercise, we start with the equation: \[7x^2 = 21\].

Here's how to isolate the variable:
  • Identify the term that contains the variable you need to solve. In this case, it鈥檚 \(7x^2\).
  • Perform an operation that will leave the variable term alone. For our equation, we divide both sides by 7, which results in \(x^2 = 3\).
  • After dividing, the variable term \(x^2\) is isolated, making it easier to solve for \(x\) in the next steps.
By isolating the variable, you're simplifying the equation, making the subsequent steps more straightforward.
simplification
Simplification involves reducing an equation or expression to its simplest form. This step often follows the isolation of the variable. It makes the equation easier to interpret and solve.

Continuing from the isolated variable: \[7x^2 = 21 \rightarrow x^2 = 3\]

Here鈥檚 what we did to simplify:
  • After dividing by 7, we were left with \(x^2 = 3\).
  • This is the simplest form of this part of the equation. It prepares us to solve for \(x\) by using another algebraic method, such as extracting square roots.
Simplification eliminates any complexity, ensuring the remaining operations are as straightforward as possible.
square roots
Extracting square roots is a common algebraic operation used to solve equations involving squared variables. Once the equation is simplified and the variable is isolated, you can solve for the variable by taking the square root of both sides.

Taking the example from before: \[x^2 = 3\]

Here鈥檚 how to solve it using square roots:
  • Apply the square root to both sides of the equation: \[x = \pm \sqrt{3}\]
  • Remember that taking the square root of a number always yields two results鈥攖he positive root and the negative root. In this case, \(x = \sqrt{3}\) and \(x = -\sqrt{3}\).
This final step gives you the complete solution for \(x\), accounting for both the positive and negative possibilities.

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Most popular questions from this chapter

Solve. $$ 25 x^{2}-4=0 $$

Replace the blanks in each equation with constants to complete the square and form a true equation. \(x^{2}+\frac{2}{3} x+\)_____\(=(x+\)_____\()^{2}\)

Solve. $$ \left(t+\frac{3}{2}\right)^{2}=\frac{7}{2} $$

Let \(g(x)=(x-2)^{2} .\) Find \(x\) such that \(g(x)=25\).

To prepare for Section \(11.2,\) review evaluating expressions and simplifying radical expressions (Sections \(1.8,10.3\) and \(10.8)\) Simplify. \([10.3],[10.8]\) $$ \sqrt{-25} $$
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