Problem 104 - Intermediate Algebra

Question

Perform each division. \(2 . 5 x - 3 . 7 \sqrt { - 2 2 . 2 5 x ^ { 2 } - 3 8 . 9 x - 1 6 . 6 5 }\)

Step-by-Step Solution

Answer

Perform operations ensuring no complex numbers arise.

Step 1: Simplify Inside the Square Root

First, simplify the expression inside the square root: \(- 22.25x^2 - 38.9x - 16.65\)

Step 2: Verify the Discriminant

Before proceeding with the division, ensure that the discriminant (\(b^2 - 4ac\)) in the expression inside the square root is not negative to avoid imaginary numbers: For the expression inside the square root, \(a = 22.25, b = 38.9, c = 16.65\). Calculate: \[b^2 - 4ac = 38.9^2 - 4(22.25)(16.65)\]Ensure that this value is positive so the division can proceed without imaginary results.

Step 3: Perform the Division Operation

With the previously simplified expression and verifying it's not complex:Using the formula: \[ \frac{2.5x}{3.7}\] for the division part of the given formula. It's important to proceed ensuring the expression under square root avoids complex numbers.

Step 4: Conclusion

Thus, the simplified form after division ensures the operation is entirely real and on constants rationality assuming every pre-condition is held valid.

Key Concepts

Understanding the Discriminant

The discriminant is a crucial component in determining the nature of a quadratic equation. It is found in the quadratic formula: \[ ax^2 + bx + c = 0 \]The discriminant itself is given by the formula: \[ b^2 - 4ac \]The value of the discriminant tells us whether the roots of the quadratic equation are real or complex (imaginary). Here’s what the discriminant reveals:

• If \( b^2 - 4ac > 0 \), the quadratic equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is exactly one real root, or a repeated root.
• If \( b^2 - 4ac < 0 \), the equation has complex roots (involves imaginary numbers).

When solving problems like the one in the exercise, it's essential to first check the discriminant of the expression inside the square root. Ensuring a non-negative value indicates that roots, and subsequent calculations, remain in the realm of real numbers. This verification step helps avoid complications with imaginary numbers in real-world applications. Remember, a real and positive discriminant simplifies further calculations.

Simplifying Square Roots

Simplifying square roots is a fundamental process that helps in reducing complexity and making calculations easier. When dealing with square roots in algebraic expressions, especially within the context of the exercise, the process involves a few key steps:

• First, simplify the expression inside the square root as much as possible. This means combining like terms and ensuring all calculations within the expression are correct.
• After verifying the expression using the discriminant, if it remains non-negative, proceed with the simplification.
• The goal is to express the square root in its simplest form, or if possible, as a rational number.

For instance, the expression under the square root \(-22.25x^2 - 38.9x - 16.65\) should be carefully analyzed and simplified using techniques like factoring or completing the square. Remember, fully simplified expressions not only make further calculations straightforward but also enhance understanding of the algebraic relationships at play. Simplifying square roots routinely aids in reaching more elegant and manageable solutions.

Division of Algebraic Expressions

Dividing algebraic expressions, such as the one in the exercise, demands attention to both the numerator and the denominator. The division process involves these steps:

• Identify both the numerator and the denominator of the expression. In this specific scenario, the division involves simplifying \( \frac{2.5x}{3.7} \).
• Check for any common factors between the numerator and denominator that may simplify the expression. This could involve reducing coefficients or simplifying any terms that both the numerator and denominator share.
• Always consider the expression within the context of real numbers, especially if prior calculations involved verifying the discriminant. This ensures avoiding division by zero or other undefined operations.

When completed correctly, division of algebraic expressions provides clear and precise solutions that maintain consistency with mathematical rules. Ensuring thorough simplification of each component before division can make calculations intuitive and less error-prone. Always verify your final result to ensure logical consistency with the initial algebraic expression.