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The Quadratic Formula is a powerful tool that allows us to find solutions to quadratic equations of the form \( ax^2 + bx + c = 0 \). The formula reads:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
It offers a systematic method to calculate the roots of any quadratic equation, regardless of whether the roots are real or complex.

• Substitute: Begin by identifying the coefficients a, b, and c from the equation, then plug these values into the formula.
• Calculate: Perform the operations inside the formula, including the discriminant which lies under the square root sign.
• Interpret: The resulting values of x are the roots of the quadratic equation, which might be one root repeated, two real roots, or two complex roots depending on the discriminant.

In the problem, we substitute \( a = \frac{7}{8} \), \( b = -\frac{3}{4} \), and \( c = \frac{5}{16} \) into the quadratic formula to find the roots of the quadratic equation. The systematic application of the quadratic formula makes it a foolproof method to tackle any quadratic problems you might face.

The discriminant is a specific part of the quadratic formula, located under the square root sign, and is denoted by \( b^2 - 4ac \). It plays a crucial role in determining the nature of the roots of a quadratic equation:

• If the discriminant is positive, the equation has two distinct real roots.
• If the discriminant is zero, the equation has exactly one real root (also known as a repeated or double root).
• If the discriminant is negative, the equation has two complex roots.

In our exercise, the discriminant calculation is \[ (-\frac{3}{4})^2 - 4 * \frac{7}{8} * \frac{5}{16} = \frac{1}{16} \], which is positive, indicating that our quadratic equation has two distinct real roots. The discriminant not only informs us about the number and type of roots but also influences the root calculation as part of the quadratic formula.

The roots of a quadratic equation are the solutions to the equation and represent the x-values where the parabola (graph of the quadratic equation) intersects the x-axis. In simpler terms, roots are the answers we get for x when the entire quadratic equation equals zero.

• Real Roots: Points where the parabola touches or crosses the x-axis.
• Complex Roots: Occur when the parabola does not intersect the x-axis at all, which happens when the discriminant is negative.

Using the Quadratic Formula we found the discriminant to be \( \frac{1}{16} \), a positive number. Thus, we have two real roots. To find the actual values:
\[ x = \frac{\frac{3}{4} \pm \sqrt{\frac{1}{16}}}{2 * \frac{7}{8}} \]
Solving this yields the roots \( x = \frac{4}{7} \) and \( x = \frac{5}{8} \), representing the points at which the quadratic equation's graph intersects the x-axis. Understanding the roots is crucial because they often represent meaningful intercepts in real-world problems modeled by quadratic equations.