Solve the equation for all values of x by completing the square. Express your answer in simplest form.

Question

x, squared, plus, 12, x, plus, 26, equals, 0
x
2
+12x+26=
0

Answer 1

To solve the equation $x^2+12x+26=0$ by completing the square, follow these steps:

1. Rearrange the terms: $$x^2+12x=-26$$

2. Complete the square:

   • Take the coefficient of $x$ (which is 12), divide it by 2 to get 6, and then square it to get $6^2=36$.
   • Add 36 to both sides of the equation: $$x^2+12x+36=-26+36$$ Simplifying the right side: $$x^2+12x+36=10$$

3. Rewrite the left side as a square: $$(x+6{)}^2=10$$

4. Take the square root of both sides: $$x+6=\pm \sqrt{10}$$

5. Isolate $x$:

   • For the positive case: $$x=-6+\sqrt{10}$$
   • For the negative case: $$x=-6-\sqrt{10}$$

6. Final solutions: $$x=-6+\sqrt{10}\text{and}x=-6-\sqrt{10}$$

Therefore, the solutions to the equation $x^2+12x+26=0$ are: $$x=-6+\sqrt{10}\text{and}x=-6-\sqrt{10}$$