Problem 62 \(2 x^{2}+1=19 ; \quad 3\)... [FREE SOLUTION]

Chapter 1: Problem 62

\(2 x^{2}+1=19 ; \quad 3\)

Short Answer

Expert verified

The solutions are x = 3 or x = -3.

Step by step solution

- Subtract 1 from both sides

The initial equation is 2x^{2} + 1 = 19 Subtract 1 from both sides to isolate the term with the variable. 2x^{2} + 1 - 1 = 19 - 1 Therefore, 2x^{2} = 18

- Divide by 2

Now, divide both sides of the equation by 2 to solve for x虏. 2x^{2}/2 = 18/2 Therefore, x^{2} = 9

- Take the square root of both sides

Take the square root of both sides of the equation to solve for x. 鈭歺^{2} = 鈭�9 Therefore, x = 卤3

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Key Concepts

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

Subtracting from both sides

When solving any equation, we often need to manipulate it to isolate the variable. One common technique is subtracting the same number from both sides of the equation. This keeps the equation balanced, meaning both sides remain equal.
In our given example, the initial equation is:
2x^{2} + 1 = 19.
Here, we subtract 1 from both sides to help isolate the term that includes the variable (in this case, \(2x^{2}\)). Performing this step, we get:
2x^{2} + 1 - 1 = 19 - 1
This simplifies to:
2x^{2} = 18.
By subtracting 1 from both sides, we've made it simpler to solve for \(x^{2}\).

Isolating the variable

After subtracting, the next goal is to isolate the variable. This means having the variable by itself on one side of the equation. In our example, we already subtracted 1, and now we can focus on \(2x^{2}\).
Our resulting equation is:
2x^{2} = 18.
Isolating the variable involves getting rid of any coefficients or other numbers attached to the variable term.
We'll do this through another step called division, which is explained next.

Dividing both sides

Next, we divide both sides of the equation by the coefficient in front of the variable term. For our example, this coefficient is 2. So, we divide both sides by 2. This step ensures that the variable \(x^{2}\) is completely isolated.
Let's see this in action:
\frac{2x^{2}}{2} = \frac{18}{2}
Simplifying, we get:
x^{2} = 9.
Dividing both sides by 2 has cleared out the coefficient and isolated \(x^{2}\).

Taking the square root

Finally, to solve for \(x\), we take the square root of both sides of the equation.
Starting with our isolated equation:
x^{2} = 9.
The square root of \(x^{2}\) is just \(x\), and the square root of 9 is 3, but we must remember that both positive and negative values could work.
Thus, we have:
鈭歺^{2} = 卤鈭�9
This results in:
x = 卤3.
Taking the square root has given us our final solutions for \(x\): both 3 and -3 are valid.

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91影视

\(2 x^{2}+1=19 ; \quad 3\)

Short Answer

Step by step solution

- Subtract 1 from both sides

- Divide by 2

- Take the square root of both sides

Key Concepts

Subtracting from both sides

Isolating the variable

Dividing both sides

Taking the square root

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