Problem 22 Solve each equation. $$3 x^{2}... [FREE SOLUTION]

Chapter 9: Problem 22

Solve each equation. $$3 x^{2}-8=46$$

Short Answer

Expert verified

The solutions are $ x = 3\sqrt{2} $ and $ x = -3\sqrt{2} $.

Step by step solution

- Isolate the quadratic term

To isolate the quadratic term, add 8 to both sides of the equation. \[ 3x^2 - 8 + 8 = 46 + 8 \] This simplifies to: \[ 3x^2 = 54 \]

- Solve for $x^2$

Next, divide both sides by 3 to solve for $x^2$. \[ \frac{3x^2}{3} = \frac{54}{3} \] This gives: \[ x^2 = 18 \]

- Solve for $x$

Take the square root of both sides to solve for $x$. \[ x = \pm \sqrt{18} \] These roots can be simplified further: \[ x = \pm \sqrt{9 \times 2} = \pm 3\sqrt{2} \]

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Key Concepts

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

Isolating Variables

In solving quadratic equations, the first step is often to isolate the quadratic term. This means we want to get the term with the variable squared by itself on one side of the equation. For example, in the equation $3x^{2} - 8 = 46$, we start by moving the constant term (-8) over to the other side. We do this by adding 8 to both sides. This gives us: \[ 3x^{2} - 8 + 8 = 46 + 8 \] which simplifies to \[ 3x^{2} = 54 \]. By isolating the variable term, we make it easier to solve for the variable in the subsequent steps.

Square Roots

After isolating the quadratic term, the next step often involves taking the square root of both sides of the equation. In our example, we first solved for $x^{2}$ by dividing both sides by 3, giving us: \[ x^{2} = 18 \]. Now, to find $x$, we take the square root of both sides. It is important to remember that the square root operation will give us two solutions, one positive and one negative: \[ x = \pm \sqrt{18} \]. These are the solutions to our quadratic equation.

Simplifying Radicals

When you take the square root of a number that is not a perfect square, like 18 in our example, you often need to simplify the radical. To simplify $ \sqrt{18} $, you factor 18 into its prime factors, which gives us \[ 18 = 9 \times 2 \]. We know that $ \sqrt{9} = 3 $, so we can rewrite this as \[ \sqrt{18} = \sqrt{9 \times 2} = 3 \sqrt{2} \]. Therefore, the solutions to our equation become \[ x = \pm 3 \sqrt{2} \].

Solving Quadratic Equations

To solve a quadratic equation like $3x^{2} - 8 = 46$, we put all these steps together. Here is a quick summary:

First, isolate the quadratic term by adding or subtracting constants (e.g., add 8 to both sides).
Next, simplify to get the variable squared term alone (e.g., divide by 3 to get $x^{2} = 18$).
Then, take the square root of both sides (e.g., $x = \pm \sqrt{18}$).
Finally, simplify the radicals if possible (e.g., $x = \pm 3 \sqrt{2}$).

These steps will help you solve most basic quadratic equations efficiently and accurately.

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91影视

Solve each equation. $$3 x^{2}-8=46$$

Short Answer

Step by step solution

- Isolate the quadratic term

- Solve for \(x^2\)

- Solve for \(x\)

Key Concepts

Isolating Variables

Square Roots

Simplifying Radicals

Solving Quadratic Equations

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