Problem 81 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 81-82: When computations are carried out on a calculator, the quadratic formula will not always give accurate results if \(b^{2}\) is large in comparison to \(a c\), because one of the roots will be close to zero and difficult to approximate. (a) Use the quadratic formula to approximate the roots of the given equation. (b) To obtain a better approximation for the root near zero, rationalize the numerator to change $$ x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \text { to } x=\frac{2 c}{-b \mp \sqrt{b^{2}-4 a c}} $$ and use the second formula. $$ x^{2}+4,500,000 x-0.96=0 $$

Step-by-Step Solution

Answer

Using the rationalized formula confirms the approximate root near zero: \(x \approx -0.0000002\).

Step 1: Identify coefficients

For the quadratic equation \(x^2 + 4,500,000 x - 0.96 = 0\), identify the coefficients: \(a = 1\), \(b = 4,500,000\), and \(c = -0.96\).

Step 2: Apply the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the coefficients: \(x = \frac{-4,500,000 \pm \sqrt{(4,500,000)^2 - 4 \times 1 \times (-0.96)}}{2 \times 1}\).

Step 3: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\): \(b^2 = (4,500,000)^2 = 20,250,000,000,000\) and \(-4ac = -4 \times 1 \times -0.96 = 3.84\). Therefore, the discriminant is \(20,250,000,000,003.84\).

Step 4: Compute the Roots Using Quadratic Formula

Calculate the roots: \(x_1 = \frac{-4,500,000 + \sqrt{20,250,000,000,003.84}}{2}\) and \(x_2 = \frac{-4,500,000 - \sqrt{20,250,000,000,003.84}}{2}\). These roots are approximated as \(x_1 \approx -0.0000002\) and \(x_2 \approx -4,500,000.0000001\).

Step 5: Rationalize the Numerator for Better Approximation

For a better approximation of the root near zero, use the rationalized formula: \(x = \frac{2c}{-b \mp \sqrt{b^2 - 4ac}}\). Substitute the values: \(x = \frac{2 \times -0.96}{-4,500,000 - \sqrt{20,250,000,000,003.84}}\).

Step 6: Calculate the Rationalized Root

Compute \(x\) using the above formula. This gives a more precise value for the root near zero. The formula simplifies to give the approximate root \(x \approx -0.0000002\), confirming the small root calculated earlier.

Key Concepts

Quadratic Formula

Quadratic equations take the form: \[ ax^2 + bx + c = 0 \]Here, the quadratic formula provides a direct method to find the roots (or solutions) of such equations:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula emerges by completing the square on the quadratic equation, offering two potential values for \(x\) due to the \(\pm\) symbol. These represent the solutions where the parabola described by the equation intersects the x-axis. The quadratic formula is efficient and provides exact results for real number coefficients, so using it requires identifying the coefficients \(a\), \(b\), and \(c\) from the given equation.
Knowing the formula allows for quick calculations, though interpreting the results sometimes needs additional refinement, especially when working with large coefficients.

Discriminant

The discriminant, represented by \(b^2 - 4ac\), is a key component within the quadratic formula. It is vital as it determines the nature of the roots of the equation:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is one real root (or a repeated root).
• If it is negative, there are no real roots, but two complex roots.

Calculating the discriminant gives insights before fully solving the equation, guiding you on what type of solutions to expect. In our original exercise, the discriminant was a substantial number, indicating two distinct real roots, but highlighted the challenge with numerical precision when roots are extremely close to zero.

Numerical Methods

Numerical methods refer to techniques used to approximate solutions when exact answers are cumbersome to compute. For quadratic equations, utilizing calculators or computer algorithms becomes essential when coefficients are large, as shown in the original exercise.
Large numbers impact precision, potentially causing significant errors, especially for roots near zero. Numerical methods help refine approximations, offering practical and more functional solutions than direct calculations offer. It involves optimizing computations, sometimes by altering formulas or simplifying complex terms to avoid arithmetic inaccuracies.

Root Approximation

Root approximation is especially critical when dealing with equations where b\(^2\) greatly overshadows \(ac\). This creates situations where one root determined by the quadratic formula is very close to zero. Such scenarios often require enhancements for accuracy.
In such cases, rationalizing the numerator provides more precise roots. By reorganizing the formula as integrated in the exercise:\[ x = \frac{2c}{-b \mp \sqrt{b^2 - 4ac}} \]The complexity of calculations decreases. The rationalized form minimizes errors brought on by subtractive cancellation, a common issue when large and small numbers interact. This step refined the approximations, delivering better precision for the root near zero, as seen in the detailed solution of the exercise.