91Ó°ÊÓ

A quadratic equation is an equation of the second degree, usually expressed in the standard form formula: \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are coefficients, where \(a eq 0\); otherwise, it would not be a quadratic equation, simply a linear equation. To solve any quadratic equation using the quadratic formula, it's essential first to rearrange it into this standard form.

• \(a\) is the coefficient of \(x^2\).
• \(b\) is the coefficient of \(x\).
• \(c\) is the constant term.

For example, in the exercise given, the initial equation \(2 + 2x - x^2 = 0\) is rearranged to \(x^2 - 2x + 2 = 0\). This makes it easier to identify \(a = -1\), \(b = 2\), and \(c = 2\). Rearranging the equation correctly sets the stage for solving it using the quadratic formula.

The roots of a quadratic equation are the solutions for \(x\) that make the equation true, usually represented by \(x_1\) and \(x_2\).

To find the roots using the quadratic formula, we follow the expression:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] The "±" symbol indicates that there are generally two roots: one by taking the plus sign, and one by taking the minus sign.

Consider the example from the exercise, substituting \(a = -1\), \(b = 2\), and \(c = 2\) into the formula:\[x = \frac{-2 \pm \sqrt{(2)^2 - 4(-1)(2)}}{2(-1)}\]This expression is simplified to find the roots \(x_1 = -1-\sqrt{3}\) and \(x_2 = -1+\sqrt{3}\). These are the values of \(x\) for which the quadratic equation equals zero. By using the quadratic formula correctly, we can calculate the roots' exact values.

The discriminant is part of the quadratic formula and is represented by \(b^2 - 4ac\). It is a key component in determining the nature of the roots of the quadratic equation.

If you compute the discriminant and find:

• It is positive, the quadratic equation has two distinct real roots.
• It is zero, the equation has exactly one real root (also known as a repeated root).
• It is negative, there are no real roots; instead, there are two complex roots.

In our specific exercise, the discriminant is calculated as \((2)^2 - 4(-1)(2) = 4 + 8 = 12\), which is positive. This positivity indicates the existence of two distinct real roots. Understanding and calculating the discriminant helps anticipate the kind of solutions one should expect from solving the quadratic equation.