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When faced with a quadratic equation like the one provided, which is in the form of ax² + bx + c = 0, the goal is to find the values of x that make the equation true. One of the most efficient methods to find these values is by utilizing the quadratic formula.

The quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), allows us to find the solutions, also known as the roots, by substituting the coefficients a, b, and c from our equation directly into the formula. Once we substitute these values, we simplify the expression to solve for x. \
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\In the example, once the values 7, 8, and 1 are substituted in as a, b, and c, simplification leads to the finding of the two roots, ensuring that each step taken is precise and clear to avoid any errors in calculations.

A key part of working with quadratic equations is understanding the discriminant, symbolized as Δ. It's calculated using the formula \(Δ = b^2 - 4ac\). The value of the discriminant can tell us much about the nature of the roots of the quadratic equation.

For instance, if Δ is positive, we get two distinct real roots. If Δ is zero, this means there is one unique real root, often referred to as a 'repeated root'. Finally, if Δ is negative, the equation does not have real roots; instead, it has two complex roots.

What does the discriminant say about our equation?

In our example, we calculated the discriminant to be 36, a positive number, indicating that there are two real and distinct roots. This important step precedes the use of the quadratic formula and sets the stage for predicting the number of possible solutions for the equation.

Upon solving a quadratic equation using the quadratic formula, the final outcomes are known as the roots or solutions of the equation. These roots can be real or complex numbers and represent the x-values where the quadratic equation intersects the x-axis on a graph.

In the given problem, by applying the quadratic formula, we find two roots: \( x = -1/7 \) and \( x = -1 \). It means that these are the points where the parabola, which is the graph of the quadratic equation, touches or crosses the x-axis.

Importance of Each Root

Each root indicates a point of equilibrium where the forces represented by the quadratic equation are in balance. Clear understanding of roots can enable students to visualize and interpret the graph's behavior, solidifying their comprehension of quadratic functions and their applications in various fields such as physics, economics, and engineering.