To solve the quadratic equation $x^2 - 10x + 19 = 0$ by completing the square, let's follow these steps:

Step 1: Move the constant term to the right side.

We start with: $x^2 - 10x + 19 = 0$ Subtract 19 from both sides: $x^2 - 10x = -19$

Step 2: Complete the square.

To complete the square, we need to find a number to add to both sides that will make the left-hand side a perfect square trinomial. Take the coefficient of $x$ (which is $-10$), divide it by 2, and square it: $\left(\frac{-10}{2}\right)^2 = 25$ Now, add 25 to both sides of the equation: $x^2 - 10x + 25 = -19 + 25$ Simplify: $x^2 - 10x + 25 = 6$

Step 3: Write the left-hand side as a square.

The left-hand side is now a perfect square trinomial: $(x - 5)^2 = 6$

Step 4: Solve for $x$.

Take the square root of both sides: $x - 5 = \pm \sqrt{6}$ Now, solve for $x$: $x = 5 \pm \sqrt{6}$

So, the two solutions are: $x = 5 + \sqrt{6}, \quad x = 5 - \sqrt{6}$

Filling in the blanks:

• The form to choose is $(x - 5)^2 = 6$.
• The solution is $x = 5 + \sqrt{6}, 5 - \sqrt{6}$.

Would you like further explanation on any step?

Relative Questions:

1. How do we derive the value needed to complete the square?
2. Can the quadratic formula be used to verify the result of completing the square?
3. Why do we add the same value to both sides of the equation when completing the square?
4. What happens if the coefficient of $x^2$ is not 1 in a quadratic equation?
5. What are other methods to solve quadratic equations besides completing the square?

Tip:

When completing the square, always remember to divide the middle coefficient by 2, then square it.