Completing the square is a powerful technique used to solve quadratic equations that cannot be easily factored. This method transforms a quadratic equation into the form \(x + a\)^2 = c, allowing the use of the square root property to find solutions. The process begins with an equation in the standard form \(ax^2 + bx + c = 0\). For completing the square, it is particularly effective when the leading coefficient \(a\) is 1 and the coefficient \(b\) is even.
To illustrate, consider the equation \(x^2 + 6x = -7\). First, rearrange it to the form \(x^2 + bx = c\) by moving the constant to the right side, resulting in \(x^2 + 6x = -7\). Next, calculate \(b/2\) and square it. Here, \(b = 6\), so \(b/2 = 3\) and \((b/2)^2 = 9\). Add this value to both sides of the equation, yielding \(x^2 + 6x + 9 = -7 + 9\), which simplifies to \(x^2 + 6x + 9 = 2\).
Now, factor the left side, which becomes \((x + 3)^2 = 2\). With the equation in the desired form, apply the square root property: take the square root of both sides, resulting in \(x + 3 = \pm \sqrt{2}\). Finally, isolate \(x\) by subtracting 3 from both sides, leading to the solutions \(x = -3 \pm \sqrt{2}\).
As a further example, consider the equation \(x^2 + 8x + 1 = 0\). Start by isolating the quadratic terms: \(x^2 + 8x = -1\). Calculate \(b/2\) where \(b = 8\), giving \(b/2 = 4\) and \((b/2)^2 = 16\). Add 16 to both sides to obtain \(x^2 + 8x + 16 = -1 + 16\), simplifying to \(x^2 + 8x + 16 = 15\). Factor the left side to get \((x + 4)^2 = 15\). Again, apply the square root property: \(x + 4 = \pm \sqrt{15}\), and isolate \(x\) to find \(x = -4 \pm \sqrt{15}\).
Completing the square not only provides a method for solving quadratic equations but also helps in understanding the properties of parabolas and their vertex forms. This technique is universally applicable to any quadratic equation, making it a valuable tool in algebra.
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