Questions: Solve by completing the square. x^2 + 12x = 19 Note: If you have a square root that does not equal a whole number, round to the nearest tenth. (1 decimal place). Write your answer as #, # (for example, 1, -2).

Solution Steps

To solve the quadratic equation by completing the square, we first move the constant term to the other side of the equation. Then, we find the value that completes the square on the left side, add it to both sides, and rewrite the left side as a perfect square trinomial. Finally, we solve for \(x\) by taking the square root of both sides and isolating \(x\).

We start with the equation: \[ x^2 + 12x = 19 \] We move the constant term to the other side: \[ x^2 + 12x - 19 = 0 \]

To complete the square, we take the coefficient of \(x\) (which is 12), divide it by 2, and square it: \[ \left(\frac{12}{2}\right)^2 = 36 \] We add this value to both sides of the equation: \[ x^2 + 12x + 36 = 19 + 36 \] This simplifies to: \[ (x + 6)^2 = 55 \]

Next, we take the square root of both sides: \[ x + 6 = \pm \sqrt{55} \] Isolating \(x\) gives us: \[ x = -6 \pm \sqrt{55} \] Calculating the two possible values: \[ x_1 = -6 + \sqrt{55} \quad \text{and} \quad x_2 = -6 - \sqrt{55} \] Approximating these values, we find: \[ x_1 \approx 1.4 \quad \text{and} \quad x_2 \approx -13.4 \]

Final Answer