91Ó°ÊÓ

Solving quadratic equations is an essential skill in algebra. A quadratic equation is typically in the form of \(ax^2 + bx + c = 0\) where \(a\), \(b\), and \(c\) are constants. To solve a quadratic equation, you often look at the discriminant first. The discriminant helps to determine how many real solutions the quadratic equation has.

To illustrate, let's consider the given equation \(x^2 + 6x + 9 = 0\). This quadratic equation can be solved by following three key steps:

• Identify coefficients \(a\), \(b\), and \(c\)
• Calculate the discriminant \(D\)
• Analyze the discriminant to determine the number of real solutions

The solutions to quadratic equations can be obtained using various methods like factoring, completing the square, or using the quadratic formula, but understanding the discriminant is fundamental to determining your approach.

The number of real solutions to a quadratic equation is determined by the value of the discriminant \(D\). The discriminant is calculated using the formula \(D = b^2 - 4ac\). Once you have the discriminant, follow these guidelines to determine the number of solutions:

• If \(D > 0\), there are two distinct real solutions
• If \(D = 0\), there is exactly one real solution
• If \(D < 0\), there are no real solutions (the solutions are complex or imaginary)

In our example equation \(x^2 + 6x + 9 = 0\), the coefficients are \(a = 1\), \(b = 6\), and \(c = 9\). When we plug these values into the discriminant formula: \(D = 6^2 - 4(1)(9) = 36 - 36 = 0\), we find that \(D = 0\). Therefore, the equation \(x^2 + 6x + 9 = 0\) has exactly one real solution.

Identifying coefficients in a quadratic equation is a crucial first step. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where

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

From our given equation \(x^2 + 6x + 9 = 0\), we can see that: \(a = 1\), \(b = 6\), \(c = 9\).

Properly identifying these coefficients allows you to calculate the discriminant accurately and thus determine the number of real solutions. Remember, recognizing these parts of the equation and their corresponding roles hands you the tools to tackle the problem systematically.