91Ó°ÊÓ

The discriminant is a crucial part of the quadratic formula, helping determine the nature and number of solutions for a given quadratic equation. It is represented by the expression within the square root sign: \( b^2 - 4ac \), where \( a \), \( b \), and \( c \) are coefficients from the standard quadratic equation format \( ax^2 + bx + c = 0 \).

The value of the discriminant can tell us much about the equation:

• If the discriminant is positive, there are two distinct real solutions.
• If it is zero, there's exactly one real solution, also known as a repeated or double root.
• If negative, it indicates there are no real solutions, but rather two complex solutions.

In our exercise, the discriminant calculated as \( 65.16 \), which is positive, confirms the presence of two distinct real solutions for the equation. Calculating the discriminant accurately is essential for predicting the nature of the quadratic's solutions.

A quadratic equation is a polynomial equation in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). It is called "quadratic" because "quad" means square, referring to the \( x^2 \) term.

Quadratic equations are fundamental in mathematics, frequently appearing in various applications like physics, engineering, and economics. Solving them involves finding the values of \( x \) that satisfy the equation. These values are also referred to as the "roots" of the equation.

In the given problem, the equation is \( 3.6x^2 + 1.8x - 4.3 = 0 \). Identifying the coefficients correctly is the first step:

• \( a = 3.6 \)
• \( b = 1.8 \)
• \( c = -4.3 \)

These coefficients are essential for applying the quadratic formula to find the solutions.

To solve quadratic equations, one powerful tool is the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula allows you to find solutions for any quadratic equation. Just plug in the coefficients \( a \), \( b \), and \( c \) into the formula, and evaluate accordingly.

The "\( \pm \)" symbol indicates there are generally two potential solutions since you will solve the equation once considering "\( + \sqrt{b^2 - 4ac} \)" and once with "\( - \sqrt{b^2 - 4ac} \)".

In our task, we substitute:

• \( b = 1.8 \)
• \( a = 3.6 \)
• \( c = -4.3 \)

into the formula. By working through the calculations:

• First calculate the discriminant \( b^2 - 4ac \) as previously discussed to be \( 65.16 \).
• Then find the square root of the discriminant: \( \sqrt{65.16} \approx 8.08 \).
• Substitute back into the quadratic formula: \[ x = \frac{-1.8 \pm 8.08}{7.2} \].

This key provides the solutions for the equation.

In many cases, especially in real-world problems, solutions may not be perfect integers or simple fractions. Hence, approximating to a desirable level of precision becomes necessary. To approximate solutions using the quadratic formula, utilize a calculator for values like square roots and divisions to get decimal results.

From our solutions:

• The first root is \( x = \frac{-1.8 + 8.08}{7.2} \approx 0.9 \).
• The second root is \( x = \frac{-1.8 - 8.08}{7.2} \approx -1.4 \).

Here, calculating each step accurately in the process ensures the reliability of the approximations. Always round your answers to the needed specificity - in this problem, it was to the nearest tenth. This precision helps provide insight while maintaining a practical approach to handling non-ideal, yet common, numerical results.