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A quadratic equation is any equation that can be written in the form \[ax^2 + bx + c = 0\]where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The term \(ax^2\) signifies it’s a second-degree polynomial, hence the name 'quadratic.' To understand quadratic equations, let's break it down:

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

In the equation \(2x^2 - 3x - 1 = 0\), the quadratic equation's standard form, we see \(a = 2\), \(b = -3\), and \(c = -1\). Recognizing these coefficients is crucial because they will be used in solving the equation.

The quadratic formula is a tool used to find the zeros (or roots) of a quadratic equation. It looks complicated at first glance, but it's very systematic. The formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here’s what each part represents:

• \(-b\): This is the negation of the coefficient \(b\).
• \(\pm\): Represents two solutions, one with addition and one with subtraction.
• \(\sqrt{b^2 - 4ac}\): The discriminant, which determines the nature of the roots.
• \(2a\): The denominator, which is twice the coefficient \(a\).

Inserting the values from our example, where \(a = 2\), \(b = -3\), and \(c = -1\), we substitute these into the quadratic formula: \[x = \frac{3 \pm \sqrt{(-3)^2 - 4(2)(-1)}}{4}\]Simplifying the expression inside the square root:\[x = \frac{3 \pm \sqrt{9 + 8}}{4}\]And further simplifying, we get:\[x = \frac{3 \pm \sqrt{17}}{4}\].

Solving quadratic equations involves several steps, and understanding each is essential. Here’s a summary using our example polynomial function: \[f(x) = 2x^2 - 3x - 1\].

Step 1: Set the function to zero

Set the function equal to zero to find its roots: \[2x^2 - 3x - 1 = 0\].

Step 2: Identify coefficients \(a\), \(b\), and \(c\)

From our equation, we recognize that:

• \(a = 2\)
• \(b = -3\)
• \(c = -1\)

Step 3: Use the quadratic formula

Apply these coefficients into the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].

Step 4: Substitute and simplify

Substitute the values \(a\), \(b\), and \(c\) into the quadratic formula, simplifying step-by-step:\[x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-1)}}{2(2)}\].Simplify inside the square root and the rest of the expression: \[x = \frac{3 \pm \sqrt{9 + 8}}{4}\].\[x = \frac{3 \pm \sqrt{17}}{4}\].

Step 5: Determine the solutions

Finally, write down the two solutions:\[x = \frac{3 + \sqrt{17}}{4}\] and \[x = \frac{3 - \sqrt{17}}{4}\].These solutions are the zeros of the function \(f(x) = 2x^2 - 3x - 1\). By following these steps meticulously, you can solve any quadratic equation.