Question

solve each equation by completing the square.
$x^2 + 6x - 19 = 0$
after completing the square, the equation is and the solution is
example: $x=3; x=5$ .

Explanation:

Step1: Move the constant term

Move -19 to the right side: \(x^2 + 6x = 19\)

Step2: Complete the square

Take half of 6, which is 3, square it to get 9. Add 9 to both sides: \(x^2 + 6x + 9 = 19 + 9\)
Simplify the left side as a perfect square: \((x + 3)^2 = 28\)

Step3: Solve for x

Take the square root of both sides: \(x + 3 = \pm\sqrt{28}\)
Simplify \(\sqrt{28}\) to \(2\sqrt{7}\): \(x + 3 = \pm 2\sqrt{7}\)
Subtract 3 from both sides: \(x = -3 \pm 2\sqrt{7}\)

Answer:

After completing the square, the equation is \((x + 3)^2 = 28\) and the solution is \(x = -3 + 2\sqrt{7}\); \(x = -3 - 2\sqrt{7}\)