Question
Fill in the steps to complete the square
X^2+12x-14=0
X^2+12x=14
X^2+12x+ blank=14+ blank
We factor and simplify to get (x+6)^2= blank
At this point, we need to blank
After this is done we will have blank= blank
Isolate the verbal by blank
Final answers are blank
X^2+12x-14=0
X^2+12x=14
X^2+12x+ blank=14+ blank
We factor and simplify to get (x+6)^2= blank
At this point, we need to blank
After this is done we will have blank= blank
Isolate the verbal by blank
Final answers are blank
Answers
GPT-4o mini
To complete the square for the equation \( x^2 + 12x - 14 = 0 \), we can follow these steps:
1. Start with the equation:
\[
x^2 + 12x - 14 = 0
\]
2. Move the constant term to the other side:
\[
x^2 + 12x = 14
\]
3. To complete the square, we need to add a specific value to both sides of the equation. The value to add is found by taking half of the coefficient of \( x \) (which is 12), squaring it, and adding it to both sides. Half of 12 is 6, and squaring it gives us \( 6^2 = 36 \):
\[
x^2 + 12x + 36 = 14 + 36
\]
4. Rewrite the left hand side as a perfect square and the right hand side:
\[
(x + 6)^2 = 50
\]
5. Now that we have the equation \( (x + 6)^2 = 50 \), we take the square root of both sides:
\[
x + 6 = \pm \sqrt{50}
\]
6. Next, isolate \( x \) by subtracting 6:
\[
x = -6 \pm \sqrt{50}
\]
Simplifying \( \sqrt{50} \) gives us \( \sqrt{25 \cdot 2} = 5\sqrt{2} \):
\[
x = -6 \pm 5\sqrt{2}
\]
So the final answers are:
\[
x = -6 + 5\sqrt{2} \quad \text{and} \quad x = -6 - 5\sqrt{2}
\]
In summary:
1. \( x^2 + 12x - 14 = 0 \)
2. \( x^2 + 12x = 14 \)
3. \( x^2 + 12x + 36 = 14 + 36 \)
4. We factor and simplify to get \( (x + 6)^2 = 50 \)
5. At this point, we need to take the square root.
6. After this is done, we will have \( x = -6 \pm 5\sqrt{2} \).
7. Isolate the variable by subtracting 6.
8. Final answers are \( x = -6 + 5\sqrt{2} \) and \( x = -6 - 5\sqrt{2} \).
1. Start with the equation:
\[
x^2 + 12x - 14 = 0
\]
2. Move the constant term to the other side:
\[
x^2 + 12x = 14
\]
3. To complete the square, we need to add a specific value to both sides of the equation. The value to add is found by taking half of the coefficient of \( x \) (which is 12), squaring it, and adding it to both sides. Half of 12 is 6, and squaring it gives us \( 6^2 = 36 \):
\[
x^2 + 12x + 36 = 14 + 36
\]
4. Rewrite the left hand side as a perfect square and the right hand side:
\[
(x + 6)^2 = 50
\]
5. Now that we have the equation \( (x + 6)^2 = 50 \), we take the square root of both sides:
\[
x + 6 = \pm \sqrt{50}
\]
6. Next, isolate \( x \) by subtracting 6:
\[
x = -6 \pm \sqrt{50}
\]
Simplifying \( \sqrt{50} \) gives us \( \sqrt{25 \cdot 2} = 5\sqrt{2} \):
\[
x = -6 \pm 5\sqrt{2}
\]
So the final answers are:
\[
x = -6 + 5\sqrt{2} \quad \text{and} \quad x = -6 - 5\sqrt{2}
\]
In summary:
1. \( x^2 + 12x - 14 = 0 \)
2. \( x^2 + 12x = 14 \)
3. \( x^2 + 12x + 36 = 14 + 36 \)
4. We factor and simplify to get \( (x + 6)^2 = 50 \)
5. At this point, we need to take the square root.
6. After this is done, we will have \( x = -6 \pm 5\sqrt{2} \).
7. Isolate the variable by subtracting 6.
8. Final answers are \( x = -6 + 5\sqrt{2} \) and \( x = -6 - 5\sqrt{2} \).