43 
0 
Completing the square is a powerful technique used in algebra to solve quadratic equations. It allows us to convert a quadratic polynomial into a perfect square trinomial, making it easier to factor and solve. This insightful method is particularly helpful when dealing with complex or irrational solutions. In this article, we will guide you through the step-by-step process of completing the square so that you can confidently solve quadratic equations like a pro.
Step 1: Start with a quadratic equation in standard form
To begin, we need a quadratic equation in the standard form, which is ax^2 + bx + c = 0. Make sure the coefficient of x^2 (a) is not zero; otherwise, it won’t be a quadratic equation. Let’s take an example equation: 3x^2 + 4x – 7 = 0.
Step 2: Isolate the x^2 and x terms
In this step, we want to isolate the terms involving x^2 and x by moving the constant term (c) to the other side of the equation. Using our example equation, we add 7 to both sides, resulting in 3x^2 + 4x = 7.
Step 3: Divide by the coefficient of x^2
To simplify the equation, we divide both sides by the coefficient of x^2 (a). Continuing with the example, we divide each term by 3, giving us x^2 + (4/3)x = 7/3.
Step 4: Adjust the coefficient of x
Now, we focus on the coefficient of x (b). To complete the square, we need to take half of this coefficient and square it. In our example, half of (4/3) is (2/3), and squaring it gives us (4/9). We add this value to both sides of the equation, making it x^2 + (4/3)x + (4/9) = 7/3 + 4/9.
Step 5: Factor the perfect square trinomial
The left side of the equation should now resemble a perfect square trinomial. The equation x^2 + (4/3)x + (4/9) can be factored into (x + 2/3)^2. On the right side, we add the fractions 7/3 + 4/9 to get a common denominator, resulting in 23/9.
Step 6: Write the equation in its completed square form
Now, the equation becomes (x + 2/3)^2 = 23/9. This step is crucial as it allows us to easily solve for x.
Step 7: Solve for x
To solve for x, we take the square root of both sides. Remember to consider both the positive and negative roots. Taking the square root of 23/9, we get x + 2/3 = ± √(23/9).
Step 8: Isolate x
Finally, we isolate the variable x by subtracting 2/3 from both sides of the equation. This gives us two solutions: x = (-2/3) ± (√23/3).
Congratulations! You have successfully completed the square and obtained the solutions to the quadratic equation. In this case, the solutions are x = (-2/3) + (√23/3) and x = (-2/3) – (√23/3).
Completing the square is an invaluable technique in algebra. It allows us to solve quadratic equations, discover their roots, and work with complex or irrational solutions. Understanding the step-by-step guide provided in this article will equip you with the necessary tools to tackle any quadratic equation efficiently. Practice completing the square, and soon enough, you’ll enhance your algebraic skills and master this useful technique.
0 | 
0