Question

Solve the following quadratic equation using the perfect square trinomial pattern: x² + 54x = -729.

Answer 1

[AI-generated answer — Bot GPT 3.5]

To solve the quadratic equation x² + 54x = -729 using the perfect square trinomial pattern, we need to rewrite the equation in the form of (x + a)² = b, where a and b are constants.

First, we move the constant term (-729) to the right side of the equation:

x² + 54x + 729 = 0

Next, we observe that the coefficient of x (54) is twice the product of the square root of the coefficient of x² (1) and the constant term (-27). This suggests that we can rewrite the equation as a perfect square trinomial.

(x + 27)² = 0

Now, we can solve for x by taking the square root of both sides of the equation:

x + 27 = ±√0

x + 27 = 0 or x + 27 = 0

Solving for x in each case:

Case 1: x + 27 = 0
x = -27

Case 2: x + 27 = 0
x = -27

Therefore, the solution to the quadratic equation x² + 54x = -729 is x = -27.

Answer 2

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

To solve the quadratic equation x² + 54x = -729 using the perfect square trinomial pattern, we need to complete the square. Here are the step-by-step instructions:

Step 1: Move the constant term to the right side of the equation:
x² + 54x + 729 = 0

Step 2: Identify the coefficients a, b, and c in the quadratic equation ax² + bx + c = 0:
In this equation, a = 1, b = 54, and c = 729.

Step 3: Divide the coefficient of x by 2 and then square it:
(b/2)² = (54/2)² = 27² = 729

Step 4: Add the squared term obtained in the previous step to both sides of the equation:
x² + 54x + 729 + 729 = 0 + 729

Simplifying:
x² + 54x + 1458 = 729

Step 5: Rewrite the left side of the equation as a perfect square trinomial:
(x + 27)² = 729

Step 6: Take the square root of both sides of the equation:
√((x + 27)²) = √(729)

Simplifying:
x + 27 = ±27

Step 7: Solve for x by subtracting 27 from both sides of the equation:
x + 27 - 27 = ±27 - 27

Simplifying:
x = -27 ± 27

Step 8: Simplify the two solutions:
x = -27 + 27 = 0
x = -27 - 27 = -54

Therefore, the solutions to the quadratic equation x² + 54x = -729 are x = 0 and x = -54.

Answer 3

[AI-generated answer — Explain Bot]

To solve the given quadratic equation x² + 54x = -729 using the perfect square trinomial pattern, we need to rewrite the equation in the form of (x + a)² = b.

Step 1: Move the constant term to the other side of the equation:
x² + 54x + 729 = 0

Step 2: Identify the coefficient of x, which is 54 in this case.
Take half of the coefficient and square it. In this case, half of 54 is 27, and 27 squared is 729.

Step 3: Rewrite the equation with the perfect square trinomial pattern:
(x + 27)² = 0

Now we have a perfect square trinomial. The equation is in the form of (x + a)² = 0, where a = 27.

Step 4: Solve for x by taking the square root of both sides:
x + 27 = 0
x = -27

So the solution to the quadratic equation x² + 54x = -729 is x = -27.