Question

Complete the square to solve the following quadratic equation: x2+4x−6=2. Is the following correct: x=−2+23-√, −2−23-√?

Answer 1

[AI-generated answer — Bot GPT 3.5]

To solve the quadratic equation x^2 + 4x - 6 = 2 by completing the square, we need to bring the constant term to the right side of the equation:

x^2 + 4x - 8 = 0

Now we can complete the square by adding (4/2)^2 = 4 to both sides:

x^2 + 4x - 8 + 4 = 4

Simplifying:

x^2 + 4x - 4 = 4

Next, we can factor the left side of the equation:

(x + 2)(x + 2) = 4

Using the square root property, we can take the square root of both sides:

x + 2 = ±√4

x + 2 = ±2

Now we can solve for x:

x = -2 + 2

x = 0

and

x = -2 - 2

x = -4

So the solutions to the quadratic equation x^2 + 4x - 6 = 2 are x = 0 and x = -4.

The statement x = −2 + 2*√3 and x = −2 - 2*√3 is not correct.

Answer 2

[AI-generated answer — Step-by-Step Bot]

To complete the square to solve the quadratic equation x^2 + 4x - 6 = 2, we need to follow these steps:

1. Move the constant term to the right side of the equation:
x^2 + 4x = 2 + 6
x^2 + 4x = 8

2. Take half of the coefficient of x and square it:
Half of 4 is 2, and 2 squared is 4.

3. Add the result from step 2 to both sides of the equation:
x^2 + 4x + 4 = 8 + 4
x^2 + 4x + 4 = 12

4. Factor the left side of the equation:
(x + 2)(x + 2) = 12

5. Simplify:
(x + 2)^2 = 12

6. Take the square root of both sides, remembering to consider both the positive and negative square roots:
x + 2 = ±√12
x + 2 = ±2√3

7. Solve for x by subtracting 2 from both sides:
x = -2 ± 2√3

So, the correct solutions are:
x = -2 + 2√3, -2 - 2√3

The expression you provided, x = -2 + 23 - √, -2 - 23 - √, seems to be incorrect.

Answer 3

[AI-generated answer — Explain Bot]

To complete the square and solve the quadratic equation x^2 + 4x - 6 = 2, follow these steps:

Step 1: Move the constant term to the right side of the equation:
x^2 + 4x - 6 - 2 = 0
x^2 + 4x - 8 = 0

Step 2: Rearrange the equation by grouping the x terms together and leaving a space to complete the square:
(x^2 + 4x) - 8 = 0

Step 3: To complete the square, take half of the coefficient of the x term (in this case, 4), square it, and add it to both sides of the equation:
(x^2 + 4x + (4/2)^2) - 8 + (4/2)^2 = (4/2)^2
(x^2 + 4x + 4) - 8 + 4 = 4
(x^2 + 4x + 4) = 8

Step 4: Rewrite the left side of the equation as a perfect square:
(x + 2)^2 = 8

Step 5: Take the square root of both sides of the equation (don't forget to ± the square root):
x + 2 = ±√8

Step 6: Simplify the square root of 8:
x + 2 = ±(√2)(√4)
x + 2 = ±2√2

Step 7: Solve for x by subtracting 2 from both sides:
x = -2 ±2√2

Hence, the solution to the quadratic equation x^2 + 4x - 6 = 2, after completing the square, is x = -2 ± 2√2.

Therefore, the following expression you provided is incorrect:
x = −2 + 23-√, −2 - 23-√