91Ó°ÊÓ

To solve a quadratic equation, we often use the quadratic formula. It is useful when factoring is difficult or impossible. The quadratic formula is: \[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).

• \( a \) is the coefficient of \( x^2 \).
• \( b \) is the coefficient of \( x \).
• \( c \) is the constant term.

The formula calculates the roots (solutions) for the quadratic equation. In this exercise, our equation \( n^2 + 33n + 270 = 0 \) can be solved by applying the formula. By substituting \( a = 1 \), \( b = 33 \), and \( c = 270 \) into the formula, you find the values of \( n \) that satisfy the equation. This method is reliable for finding the roots of any quadratic equation.

The discriminant is a part of the quadratic formula under the square root: \( b^2 - 4ac \). It determines the nature of the roots.

• If \( b^2 - 4ac > 0 \), you get two distinct real solutions.
• If \( b^2 - 4ac = 0 \), there is exactly one real solution, a repeated root.
• If \( b^2 - 4ac < 0 \), there are no real solutions but two complex solutions.

In our problem, the discriminant \( 33^2 - 4 \times 1 \times 270 = 9 \) was calculated. The results indicate a positive and perfect square value, showing there are two distinct and real solutions. Understanding this concept helps anticipate the type of answers before fully solving the equation.

When solving quadratic equations, a rational solution is one that can be expressed as a ratio of two integers (like \( \frac{p}{q} \)). These are easier to interpret and work with than irrational or complex solutions. In the exercise \( n^2 + 33n + 270 = 0 \), we found the discriminant was a perfect square (9), which often leads to rational solutions. Solving the equation, we got \( n = -15 \) and \( n = -18 \). These are both rational numbers, as you can write them as fractions (\( \frac{-30}{2} \) and \( \frac{-36}{2} \)). Knowing the solutions are rational helps confirm our understanding and the equation's correct solution. This is valuable in various practical and academic situations where simpler numerical values are needed.