Solve x2=1625

Question

by inspection. There are two real solutions. Enter the lesser number first. Leave the answers in simplest fraction form. (1 point)

Answer 1

To solve the equation \( x^2 = 1625 \) by inspection, we take the square root of both sides:

\[ x = \pm \sqrt{1625} \]

Next, we simplify \( \sqrt{1625} \). We can factor 1625 into its prime factors:

\[ 1625 = 25 \times 65 = 25 \times 5 \times 13 = 5^2 \times 5 \times 13 = 5^3 \times 13 \]

Now we can simplify \( \sqrt{1625} \):

\[ \sqrt{1625} = \sqrt{25 \times 65} = \sqrt{25} \times \sqrt{65} = 5 \sqrt{65} \]

Thus, the solutions for \( x \) are:

\[ x = \pm 5\sqrt{65} \]

The two real solutions are \( -5\sqrt{65} \) and \( 5\sqrt{65} \).

To ensure these answers are in simplest fraction form:

• As \( \sqrt{65} \) is not a perfect square, we can leave it as it is in radical form.

Thus, the final solutions, ordered with the lesser number first, are:

\[ -5\sqrt{65}, 5\sqrt{65} \]

Answer 2

Solve the following quadratic equation using square roots: (x+3)2=36
.(1 point)

Answer 3

To solve the quadratic equation \( (x + 3)^2 = 36 \) using square roots, follow these steps:

1. Take the square root of both sides:

   \[ x + 3 = \pm \sqrt{36} \]

2. Simplify:

   \[ x + 3 = \pm 6 \]

3. Solve for \( x \) by considering both the positive and negative cases.

   • Case 1: \[ x + 3 = 6 \] Subtract 3 from both sides: \[ x = 6 - 3 = 3 \]

   • Case 2: \[ x + 3 = -6 \] Subtract 3 from both sides: \[ x = -6 - 3 = -9 \]

4. Final solutions:

The solutions to the equation \( (x + 3)^2 = 36 \) are:

\[ x = 3 \quad \text{and} \quad x = -9 \]

Answer 4

Solve the following quadratic equation using square roots. Round to the nearest hundredth if necessary: (x−6)2=51
.(1 point)

Answer 5

To solve the quadratic equation \( (x - 6)^2 = 51 \) using square roots, follow these steps:

1. Take the square root of both sides: \[ x - 6 = \pm \sqrt{51} \]

2. Simplify: \[ x - 6 = \sqrt{51} \quad \text{or} \quad x - 6 = -\sqrt{51} \]

3. Solve for \( x \) by considering both the positive and negative cases.

   • Case 1: \[ x - 6 = \sqrt{51} \] Add 6 to both sides: \[ x = 6 + \sqrt{51} \]

   • Case 2: \[ x - 6 = -\sqrt{51} \] Add 6 to both sides: \[ x = 6 - \sqrt{51} \]

4. Evaluate the square root and round to the nearest hundredth: \[ \sqrt{51} \approx 7.14 \]

   • For Case 1: \[ x = 6 + \sqrt{51} \approx 6 + 7.14 \approx 13.14 \]
   • For Case 2: \[ x = 6 - \sqrt{51} \approx 6 - 7.14 \approx -1.14 \]

5. Final solutions:

The solutions to the equation \( (x - 6)^2 = 51 \) are approximately:

\[ x \approx 13.14 \quad \text{and} \quad x \approx -1.14 \]

Answer 6

thats rong it wants you to find the total and round it if needed

Answer 7

Apologies for the misunderstanding. Let's combine the solutions correctly and provide the total.

Given the solutions we derived from the equation \( (x - 6)^2 = 51 \):

1. The two solutions are:

   • \( x_1 = 6 + \sqrt{51} \)
   • \( x_2 = 6 - \sqrt{51} \)

2. Calculating the values:

   • We previously approximated \( \sqrt{51} \approx 7.14 \):
     • \( x_1 \approx 6 + 7.14 = 13.14 \)
     • \( x_2 \approx 6 - 7.14 = -1.14 \)

3. Finding the total of the two solutions: \[ \text{Total} = x_1 + x_2 = (6 + \sqrt{51}) + (6 - \sqrt{51}) \]

   • The \(\sqrt{51}\) cancels out: \[ \text{Total} = 6 + 6 = 12 \]

So, the total of the two solutions is \( \boxed{12} \).

Thank you for your clarification!

Answer 8

its not 12
this is the eqation (x−6)2=51
Solve the following quadratic equation using square roots
Round to the nearest hundredth if necessary

Answer 9

I apologize for the confusion in the previous responses. Let's solve the equation \( (x - 6)^2 = 51 \) correctly this time and find the actual values.

Steps to solve:

1. Take the square root of both sides: \[ x - 6 = \pm \sqrt{51} \]

2. Calculate \( \sqrt{51} \):

   • First, we find the approximate value of \( \sqrt{51} \): \[ \sqrt{51} \approx 7.14 \]

3. Set up the two equations:

   • Positive case: \[ x - 6 = \sqrt{51} \] \[ x = 6 + \sqrt{51} \approx 6 + 7.14 \approx 13.14 \]

   • Negative case: \[ x - 6 = -\sqrt{51} \] \[ x = 6 - \sqrt{51} \approx 6 - 7.14 \approx -1.14 \]

4. Final solutions:

   • \( x_1 \approx 13.14 \)
   • \( x_2 \approx -1.14 \)

Total of the solutions:

Next, we will find the total of these solutions. \[ \text{Total} = 13.14 + (-1.14) = 12.00 \]

Rounding to the nearest hundredth:

Since this calculation results in a whole number, the final answer remains: \[ \text{Total} = 12.00 \]

So, after verifying everything step by step, the total of the two solutions is 12.00. Thank you for your patience!

Answer 10

still wrong

Answer 11

I apologize for any confusion in my previous responses. Let's solve the equation step-by-step and find the correct total while being mindful of rounding as needed at each step.

Given Equation

\[ (x - 6)^2 = 51 \]

Step 1: Take the square root of both sides

\[ x - 6 = \pm \sqrt{51} \]

Step 2: Calculate \(\sqrt{51}\)

To find \( \sqrt{51} \):

\[ \sqrt{51} \approx 7.1414 \]

Step 3: Solve for \(x\)

Now, we’ll consider both cases (positive and negative):

1. Positive Case: \[ x - 6 = \sqrt{51} \implies x = 6 + \sqrt{51} \approx 6 + 7.1414 \approx 13.1414 \]

2. Negative Case: \[ x - 6 = -\sqrt{51} \implies x = 6 - \sqrt{51} \approx 6 - 7.1414 \approx -1.1414 \]

Step 4: Round to the nearest hundredth

• For \( x_1 \): \[ x_1 \approx 13.14 \]
• For \( x_2 \): \[ x_2 \approx -1.14 \]

Step 5: Find the total of the solutions

Now, we need to sum the solutions:

\[ \text{Total} = 13.14 + (-1.14) = 12.00 \]

Conclusion

The total of the solutions is indeed \( \mathbf{12.00} \).

If you're looking for individual answers rather than their total, the two approximate answers are:

• \( x_1 \approx 13.14 \)
• \( x_2 \approx -1.14 \)

The total remains 12.00 when summarizing both solutions.

If you meant something else by "finding the total," please clarify so I can assist you better! Thank you for your understanding.