To solve the quadratic equation [tex]\( x^2 + 14x = -69 \)[/tex] by completing the square, we will follow these steps:
1. Move the constant term to the other side of the equation:
[tex]\[
x^2 + 14x + 69 = 0
\][/tex]
2. Complete the square:
We need to add and subtract the same value inside the equation to transform it into a perfect square trinomial.
- Take the coefficient of [tex]\( x \)[/tex], which is 14, and divide it by 2, then square the result:
[tex]\[
\left( \frac{14}{2} \right)^2 = 7^2 = 49
\][/tex]
- Add and subtract 49 to the left-hand side of the equation:
[tex]\[
x^2 + 14x + 49 - 49 + 69 = 0
\][/tex]
- This simplifies to:
[tex]\[
(x + 7)^2 - 49 + 69 = 0
\][/tex]
- Further simplification results in:
[tex]\[
(x + 7)^2 + 20 = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex] by isolating the square term:
[tex]\[
(x + 7)^2 = -20
\][/tex]
4. Take the square root of both sides:
[tex]\[
x + 7 = \pm \sqrt{-20}
\][/tex]
We know that the square root of a negative number involves an imaginary unit [tex]\( i \)[/tex], where [tex]\( i = \sqrt{-1} \)[/tex]. Thus:
[tex]\[
\sqrt{-20} = \sqrt{-1 \cdot 20} = 2i\sqrt{5}
\][/tex]
5. Express the solutions:
[tex]\[
x + 7 = \pm 2i\sqrt{5}
\][/tex]
- Subtract 7 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
x = -7 \pm 2i\sqrt{5}
\][/tex]
Thus, the solutions to the quadratic equation [tex]\( x^2 + 14x = -69 \)[/tex] are:
[tex]\[
x_1 = -7 + 2i\sqrt{5} \quad \text{and} \quad x_2 = -7 - 2i\sqrt{5}
\][/tex]
When expressed numerically, these solutions are approximately:
[tex]\[
((-7 + 4.47213595499958i), (-7 - 4.47213595499958i))
\][/tex]
