Completing the square is a useful way to solve quadratic equations. It also connects to something called the quadratic formula.
When you have a quadratic equation like this:
[ ax^2 + bx + c = 0 ]
You can complete the square to change it into this form:
[ (x - p)^2 = q ]
Here’s how it works:
How It Relates:
Rearranging the Equation: Completing the square helps organize the equation into a perfect square. This makes it easier to find solutions (or roots).
Getting to the Quadratic Formula: By using this method, you can derive the quadratic formula:
[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
In this formula, you are solving for ( x ) after isolating the square part.
Example:
Let’s look at the equation:
[ x^2 + 6x + 8 = 0 ]
If we complete the square, we get:
[ (x + 3)^2 = 1 ]
From here, we can find the values of ( x ):
[ x = -3 \pm 1 ]
So, the solutions are:
[ x = -2 \quad \text{and} \quad x = -4 ]
You can also find these answers using the quadratic formula!
Completing the square is a useful way to solve quadratic equations. It also connects to something called the quadratic formula.
When you have a quadratic equation like this:
[ ax^2 + bx + c = 0 ]
You can complete the square to change it into this form:
[ (x - p)^2 = q ]
Here’s how it works:
How It Relates:
Rearranging the Equation: Completing the square helps organize the equation into a perfect square. This makes it easier to find solutions (or roots).
Getting to the Quadratic Formula: By using this method, you can derive the quadratic formula:
[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
In this formula, you are solving for ( x ) after isolating the square part.
Example:
Let’s look at the equation:
[ x^2 + 6x + 8 = 0 ]
If we complete the square, we get:
[ (x + 3)^2 = 1 ]
From here, we can find the values of ( x ):
[ x = -3 \pm 1 ]
So, the solutions are:
[ x = -2 \quad \text{and} \quad x = -4 ]
You can also find these answers using the quadratic formula!