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In the context of quadratic equations, the discriminant is a powerful tool that helps us determine the nature of the roots without solving the equation completely. Every quadratic equation is in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The discriminant \( D \) is given by the formula:

• \( D = b^2 - 4ac \)

The discriminant reveals the nature of the solutions as follows:

• If \( D > 0 \), the quadratic equation has two distinct real roots.
• If \( D = 0 \), there is exactly one real root, which is also called a repeated or double root.
• If \( D < 0 \), there are no real roots, but two complex roots.

In our example, we calculated the discriminant for the equation \( 3x^2 - 5x - 2 = 0 \). By substituting \( a = 3 \), \( b = -5 \), and \( c = -2 \) into the formula, we found that the discriminant is 49, which is positive. Therefore, our equation has two distinct real roots.

The quadratic formula is a crucial tool for finding solutions of quadratic equations. This formula allows you to directly calculate the roots without the need to factor the equation. The formula is expressed as:

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

Here, \( b^2 - 4ac \) is the discriminant, which you've calculated previously. The \( \pm \) symbol indicates that there are two possible solutions: one with addition and one with subtraction. Each solution corresponds to the two roots of the equation.
For the quadratic equation \( 3x^2 - 5x - 2 = 0 \), by substituting \( a = 3 \), \( b = -5 \), and the discriminant \( \sqrt{49} = 7 \) into the formula, we can find the roots:

• \( x_1 = \frac{5 + 7}{6} = 2 \)
• \( x_2 = \frac{5 - 7}{6} = -\frac{1}{3} \)

These calculated values \( x = 2 \) and \( x = -\frac{1}{3} \) are the solutions to the quadratic equation.

Solving quadratic equations is a fundamental aspect of algebra, often involving methods like factoring, completing the square, or the quadratic formula. It is through solving these equations that we uncover the values of \( x \) that satisfy \( ax^2 + bx + c = 0 \). Here’s a simple process to tackle a quadratic equation using the quadratic formula:

• Identify \( a \), \( b \), and \( c \) from the given equation.
• Calculate the discriminant \( b^2 - 4ac \).
• Apply the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Solve for \( x \) by computing both the addition and subtraction parts.

Once you have your solutions, it's wise to verify them by substituting the values back into the original equation. For the equation \( 3x^2 - 5x - 2 = 0 \), the roots \( x = 2 \) and \( x = -\frac{1}{3} \) were checked and found correct. This checking step ensures your solutions are accurate and makes you confident in your problem-solving approach.