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The quadratic formula is a powerful tool used to find the solutions of quadratic equations. A quadratic equation is any equation of the form \( ax^2 + bx + c = 0 \). The quadratic formula is expressed as:

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

This formula allows you to calculate the values of \( x \) which make the quadratic equation true, known as the roots of the equation. The quadratic formula can be applied to any quadratic equation, no matter the values of \( a \), \( b \), and \( c \), as long as \( a eq 0 \).

By using the quadratic formula, you can find the two possible values for \( x \) by computing \( -b + \sqrt{b^2 - 4ac} \) and \( -b - \sqrt{b^2 - 4ac} \). Each solution provides a point where the parabola represented by the quadratic equation touches or crosses the x-axis.

The discriminant is a key component of the quadratic formula, represented by \( \Delta = b^2 - 4ac \). It provides valuable information about the nature of the roots of the quadratic equation:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has exactly one real root, meaning the parabola touches the x-axis at one point.
• If \( \Delta < 0 \), the quadratic equation has two complex roots, meaning the parabola does not touch the x-axis.

For the example equation \( 3x^2 + 8x + 2 = 0 \), the discriminant is calculated as \( \Delta = 64 - 24 = 40 \). Since \( 40 \) is greater than zero, the quadratic equation has two distinct real roots. This means the parabola intersects the x-axis at two different points.

The roots of a quadratic equation refer to the values of \( x \) that satisfy the equation. In other words, they are the solutions where the equation equals zero. Roots can be found using various methods, including:

• Factoring - expressing the quadratic as a product of two binomials.
• Completing the square - rewriting the equation in a form that reveals the roots.
• Using the quadratic formula - a universal method applicable to all quadratic equations.

In our example with the equation \( 3x^2 + 8x + 2 = 0 \), the roots were determined using the quadratic formula. After calculating the discriminant and substituting back into the quadratic formula, we found the solutions:

• \( x = \frac{-4 + \sqrt{10}}{3} \)
• \( x = \frac{-4 - \sqrt{10}}{3} \)

These roots tell us the x-values where the graph of the quadratic equation intersects the x-axis. Each root represents a solution to the equation, confirming the points where the equation holds true.