Solve x2 + 12x = –20 by completing the square.

The Correct Answer and Explanation is:

To solve the quadratic equation (x^2 + 12x = -20) by completing the square, we will follow a step-by-step method that involves transforming the quadratic expression into a perfect square trinomial.

Step 1: Rearrange the equation

We start by moving the constant term on the right side of the equation to the left side:

[
x^2 + 12x = -20
]

[
x^2 + 12x + _ = -20 + _
]

The blank ((_ )) will be filled with the value needed to complete the square. This value is determined in the next step.

Step 2: Complete the square

To complete the square, we need to take half of the coefficient of (x) (the linear term) and square it. In this case, the coefficient of (x) is 12. Half of 12 is 6, and squaring it gives (6^2 = 36).

Add 36 to both sides of the equation:

[
x^2 + 12x + 36 = -20 + 36
]

[
x^2 + 12x + 36 = 16
]

Step 3: Write the left-hand side as a perfect square trinomial

The left side of the equation is now a perfect square trinomial, which can be factored as:

[
(x + 6)^2 = 16
]

Step 4: Solve for (x)

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

[
x + 6 = \pm \sqrt{16}
]

[
x + 6 = \pm 4
]

Step 5: Isolate (x)

Now, isolate (x) by subtracting 6 from both sides:

For the positive square root:

[
x + 6 = 4 \quad \Rightarrow \quad x = 4 – 6 = -2
]

For the negative square root:

[
x + 6 = -4 \quad \Rightarrow \quad x = -4 – 6 = -10
]

Final Answer:

The solutions to the equation (x^2 + 12x = -20) are:

[
x = -2 \quad \text{or} \quad x = -10
]

Explanation:

By completing the square, we transformed the equation into a form where the left side is a perfect square trinomial, ((x + 6)^2), and the right side is a constant, 16. This allowed us to easily solve for (x) by taking the square root of both sides. Completing the square is a powerful technique for solving quadratic equations when factoring is not straightforward.