What is Completing the Square in Quadratic Equations?
Quick Answer
Completing the square is a method used to solve quadratic equations by rewriting them in the form (x - a)² = b. This technique helps find the roots of the equation and is useful in various mathematical applications.
Completing the square is a powerful algebraic technique used to solve quadratic equations of the form ax² + bx + c = 0. This method allows us to convert a quadratic equation into a perfect square trinomial, making it easier to find the roots of the equation.
To illustrate this, consider the quadratic equation: x² - 22x + 1 = 15. The first step in completing the square is to move the constant term to the other side of the equation. This gives us:
x² - 22x + 1 - 15 = 0, which simplifies to x² - 22x - 14 = 0.
Next, we want to isolate the x² and x terms:
x² - 22x = 14.
Now, to complete the square, we take half of the coefficient of x (which is -22), square it, and add it to both sides of the equation. Half of -22 is -11, and squaring it gives us:
(-11)² = 121.
Adding 121 to both sides, we have:
x² - 22x + 121 = 14 + 121.
This simplifies to:
(x - 11)² = 135.
At this point, we can take the square root of both sides to solve for x:
x - 11 = ±√135.
Breaking down √135, we find:
√135 = √(9 × 15) = 3√15.
Thus, we arrive at:
x - 11 = ±3√15.
This leads us to the final solutions:
x = 11 + 3√15 or x = 11 - 3√15.
Completing the square not only helps in solving quadratic equations but also plays a vital role in graphing parabolas and understanding their properties. When you rewrite a quadratic in the form (x - a)² = b, it provides insights into the vertex of the parabola and its transformations.
In real-world applications, completing the square can be used in optimization problems, physics, and engineering, where understanding the maximum or minimum values of quadratic functions is essential. Overall, mastering this technique will enhance your algebra skills and prepare you for more advanced mathematical concepts.
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