How to Solve a Quadratic Equation x² + 10x + 24 by Completing the Square

To solve the quadratic equation x² + 10x + 24 = 0 by completing the square, follow these steps:

1. Move the constant term to the other side:

Start by rearranging the equation:

x² + 10x = -24

2. Complete the square:

To complete the square, we need to add the square of half the coefficient of x to both sides of the equation. The coefficient of x is 10, so half of that is 5, and squaring it gives us 25.

Add 25 to both sides:

x² + 10x + 25 = 1

3. Rewrite the left side as a square:

The left side can now be factored as:

(x + 5)² = 1

4. Take the square root of both sides:

Now, take the square root of both sides, remembering to consider both the positive and negative roots:

x + 5 = ±1

5. Isolate x:

We solve for x by isolating it:

1. For the positive root: x + 5 = 1
   ightarrow x = 1 – 5
   ightarrow x = -4
2. For the negative root: x + 5 = -1
   ightarrow x = -1 – 5
   ightarrow x = -6

The solutions to the quadratic equation x² + 10x + 24 = 0 are:

x = -4 and x = -6

Completing the square is a useful technique, especially when you want to find vertex form or solve quadratics that may not be easily factorable using traditional methods.