Completing the Square: 3-2x-x^2
Completing the square is a powerful technique used to solve quadratic equations of the form ax^2 + bx + c = 0. In this article, we will focus on completing the square for the quadratic equation 3-2x-x^2.
What is Completing the Square?
Completing the square is a method used to transform a quadratic equation into the form (x + d)^2 = e, where d and e are constants. This form allows us to easily solve for x.
The Quadratic Equation: 3-2x-x^2
The quadratic equation we want to complete the square for is 3-2x-x^2. To start, we can rewrite the equation in the standard form:
-x^2 - 2x + 3 = 0
Step 1: Move the Constant Term to the Right
Our first step is to move the constant term, 3, to the right-hand side of the equation:
-x^2 - 2x = -3
Step 2: Add (b/2)^2 to Both Sides
Next, we add (b/2)^2 to both sides of the equation, where b is the coefficient of the x term. In this case, b = -2, so we add ((-2)/2)^2 = 1 to both sides:
-x^2 - 2x + 1 = -3 + 1
This simplifies to:
-x^2 - 2x + 1 = -2
Step 3: Factor the Left-Hand Side
Now, we can factor the left-hand side of the equation:
-(x^2 + 2x - 1) = -2
Step 4: Rewrite in Completed Square Form
Finally, we can rewrite the equation in completed square form:
-(x + 1)^2 = -2
Solving for x
To solve for x, we can take the square root of both sides of the equation:
x + 1 = ±√2
Subtracting 1 from both sides gives us the final solutions:
x = -1 ± √2
And that's it! We have successfully completed the square for the quadratic equation 3-2x-x^2.
Conclusion
Completing the square is a powerful technique for solving quadratic equations. By following the steps outlined above, we can transform any quadratic equation into completed square form, making it easy to solve for x. In this article, we applied the technique to the quadratic equation 3-2x-x^2, arriving at the solutions x = -1 ± √2.
