91Ó°ÊÓ

Completing the square is a method used for solving quadratic equations. It involves converting the equation into a perfect square trinomial, which can be easily solved for the variable. The main goal of this technique is to express the quadratic equation in the form

• y = (x-a)^2 = b

where both 'a' and 'b' are constants.
First, ensure that the coefficient of the quadratic term \(x^2\) is 1. If it isn't, factor it out from all terms.
Next, to "complete the square," add and subtract the square of half the coefficient of the linear term (the x-term) from both sides of the equation.
This forms a perfect square trinomial on one side, which can be neatly factorized.
For example, in the equation \(x^2 - 10x - 15 = 0\), you identify \(-10\) as the coefficient of the x-term.
Take half of it, say \(-5\), square it to get \(25\), then add and subtract \(25\) inside the equation:

• \(x^2 - 10x + 25 - 25 - 15 = 0\) becomes \((x - 5)^2 - 40 = 0\)

Combining terms gives \((x-5)^2 = 40\). You can now solve for x by taking the square root of each side and re-arranging terms.

The quadratic formula is a universal method for finding the solutions of any quadratic equation of the form \(ax^2 + bx + c = 0\).
This formula provides an efficient and consistent way to determine the roots and is given by:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Rather than completing the square for each problem, using the quadratic formula ensures a straightforward application to solve the equation.
Consider the example \(x^2 - 10x - 15 = 0\) where \(a = 1\), \(b = -10\), and \(c = -15\):

• Plug these values into the formula to get \(x = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot (-15)}}{2 \cdot 1}\)
• Simplify inside the square root to \(\sqrt{100 + 60} = \sqrt{160}\)

Further simplifying gives \(x = \frac{10 \pm \sqrt{160}}{2}\). The square root of 160 can be broken into \(\sqrt{40 \times 4}\) and further simplifies: \(x = 5 \pm \sqrt{40}\).
This matches the solutions obtained by completing the square, confirming the accuracy and reliability of the quadratic formula.

The roots of quadratic equations are the solutions, or the values of x, that satisfy the equation \(ax^2 + bx + c = 0\). Understanding the nature of these roots provides insight into the behavior of the quadratic function.
There are three potential outcomes:

• **Two Real Roots**: When the discriminant \(b^2 - 4ac\) is positive, the quadratic equation has two distinct real solutions.
• **One Real Root**: If the discriminant equals zero, the quadratic equation has exactly one real solution, also known as a repeated or double root.
• **Two Complex Roots**: If the discriminant is negative, the equation has two complex solutions, which occur as conjugate pairs.

In the example \(x^2 - 10x - 15 = 0\), the discriminant is \(100 - 4(-15) = 160\), which is positive, indicating two real and distinct roots.
Recognizing the type of roots tells you about the graph of the quadratic function: two real roots mean the parabola crosses the x-axis at two points.
Such analysis not only helps in solving equations but also in graphically understanding quadratic functions.