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The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula provides a straightforward method to find the solutions, or "roots," of a quadratic equation without needing to factor it. Here is the quadratic formula in its standard form: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]When using the quadratic formula:

• Identify the coefficients \(a\), \(b\), and \(c\) in the quadratic equation.
• Substitute these values into the formula.
• Remember that \( \pm \) means you will likely find two solutions: one with a plus and one with a minus.

The quadratic formula can be applied to every quadratic equation, making it an indispensable tool in algebra. This method is especially helpful when the quadratic does not factor neatly. Once you substitute the values correctly, the formula will automatically solve for the potential values of \( x \).

The discriminant is a component of the quadratic formula within the square root: \( b^2 - 4ac \). It plays a critical role in determining the number and type of roots a quadratic equation has. Calculating the discriminant can tell us whether the roots are real or imaginary, and if they are distinct or repeated.Here's what the discriminant indicates:

• If \( b^2 - 4ac > 0 \), the quadratic equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is exactly one real root, also known as a repeated or double root.
• If \( b^2 - 4ac < 0 \), the equation has no real roots; instead, it has two complex roots.

In our original exercise, the discriminant calculated as \( 196 \), which is greater than zero. This confirms that the equation \( 5x^2 - 6x - 8 = 0 \) has two distinct real roots, which aligns with the solved roots.

The roots of a quadratic equation are the values of \( x \) that make the equation true (i.e., when the equation equals zero). Finding the roots is the ultimate goal of solving a quadratic equation since they indicate where the function graph intersects the x-axis.Based on the quadratic formula, the roots obtained for the equation \( 5x^2 - 6x - 8 = 0 \) are:

• The first root: \( x = 2 \)
• The second root: \( x = -\frac{4}{5} \)

These roots are calculated by substituting back the simplified expressions from the quadratic formula, accounting for both the addition and subtraction of the square root component. This process leads us to two solutions because the quadratic equation represents a parabola, often intersecting the x-axis at two points except when the discriminant indicates otherwise.