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The quadratic formula is a fundamental tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). It's especially useful when factoring is difficult or impossible. The formula is given by:

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

This formula allows you to find the roots (solutions) of the quadratic equation by substituting the coefficients \( a \), \( b \), and \( c \) into it. The \( \pm \) symbol indicates that there will typically be two solutions: one using the plus sign and one using the minus sign. These represent the places where the parabola intersects the x-axis on a graph. Understanding this key formula is crucial for tackling quadratic problems. When using the quadratic formula, always ensure to handle the values under the square root carefully, as this can determine whether the solutions are real or complex numbers.

The discriminant plays a crucial role in understanding the nature and number of solutions of a quadratic equation. It's the part of the quadratic formula under the square root, \( b^2 - 4ac \).

• If the discriminant is positive (> 0), there are two distinct real solutions.
• If the discriminant is zero, there is exactly one real solution (or a repeated root).
• If the discriminant is negative (< 0), the solutions are complex (not real).

In our exercise, the discriminant was computed as:

• \( 0.01 - (-0.32) = 0.33 \)

This value is positive, indicating that the quadratic equation has two distinct real roots. The concept of the discriminant not only helps in solving quadratic equations but also gives insight into the graph of the function represented by the equation (a parabola), showing whether and how it intersects the x-axis.

Once the solutions to a quadratic equation are found using the quadratic formula, they may need approximation, especially if they are irrational numbers. Approximating solutions involves rounding them to a specified precision. In our scenario, we needed to round to the nearest hundredth.

• For example, if a calculated root is \( 0.119 \), we round it to \( 0.12 \).
• Similarly, a root of \( -0.169 \) is rounded to \( -0.17 \).

Rounding ensures results are presented clearly and is often necessary in practical applications, such as measurements where exact values aren't feasible. It's a straightforward step once solutions are calculated but essential for presenting neat and precise results. Remember, even though rounding simplifies the expressions, it's crucial to maintain as much precision as needed when initially performing calculations to avoid errors.