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Chapter 3: Problem 9

Solve. $$3 x^{2}=21$$

Short Answer

Expert verified
The solutions are \( x = \sqrt{7} \) and \( x = -\sqrt{7} \).

Step by step solution

01

Isolate the Quadratic Term

To isolate the quadratic term, divide both sides of the equation by 3. \[ x^2 = \frac{21}{3} \] This simplifies to \[ x^2 = 7 \].
02

Take the Square Root of Both Sides

To solve for \( x \), take the square root of both sides. Remember that taking the square root of a number results in both a positive and negative value. Thus, \[ x = \pm \sqrt{7} \].

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Key Concepts

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

Solving Quadratics
Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). The solutions to these equations are the values of \(x\) that make the equation true.
Quadratics can be solved using various methods like factoring, completing the square, or the quadratic formula. In our example, we used the square root method as it suited the simplified form of the equation after initial isolation.
This process involves reducing the quadratic equation into a simpler form, where a straightforward application of square roots helps in finding the solutions.
Square Root Method
The square root method is a powerful tool for solving quadratic equations, especially when linear terms (terms with just \(x\)) are not present. Here's how it works:
  • Start by having your equation in the form \(ax^2 = c\).
  • Divide both sides by \(a\) to isolate \(x^2\).
  • Take the square root of both sides of the equation. Remember, taking a square root will result in two solutions: a positive and a negative root.

In our example, after isolating the quadratic term, we took the square root of both sides of the equation \(x^2 = 7\). This yielded two solutions: \(x = \sqrt{7}\) and \(x = -\sqrt{7}\).
Isolate the Variable
Isolating the variable is an essential step in solving equations. It involves manipulating the equation to have the variable on one side and all other terms on the other side.
For quadratic equations, this often means moving all terms except the quadratic term \(ax^2\) to the opposite side of the equation.
In our given problem, this step was about dividing both sides of the initial equation \(3x^2 = 21\) by 3, which simplified it to \(x^2 = 7\).
This made it easier to apply the square root method directly and find the solutions for \(x\). Always ensure to simplify as much as possible before applying further solving techniques.

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Most popular questions from this chapter

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