Question

x^2-14x+19=0 by completing the square

Answer 1

To solve the equation \( x^2 - 14x + 19 = 0 \) by completing the square, follow these steps:

1. **Rearrange the equation**:
Start with the equation:
\[
x^2 - 14x + 19 = 0
\]
Move the constant to the other side:
\[
x^2 - 14x = -19
\]

2. **Complete the square**:
To complete the square, take the coefficient of \( x \), which is \(-14\), divide it by 2, and square the result:
\[
\left(\frac{-14}{2}\right)^2 = (-7)^2 = 49
\]
Add this square to both sides of the equation:
\[
x^2 - 14x + 49 = -19 + 49
\]
Simplifying the right side, we get:
\[
x^2 - 14x + 49 = 30
\]

3. **Rewrite the left side as a square**:
The left side can now be expressed as:
\[
(x - 7)^2 = 30
\]

4. **Solve for \( x \)**:
Take the square root of both sides:
\[
x - 7 = \pm \sqrt{30}
\]
Now, solve for \( x \):
\[
x = 7 \pm \sqrt{30}
\]

Thus, the solutions to the equation \( x^2 - 14x + 19 = 0 \) are:
\[
x = 7 + \sqrt{30} \quad \text{and} \quad x = 7 - \sqrt{30}
\]