Problem 47 - Precalculus: Functions and Graphs

Question

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

Use the rationalized formula for a small root: \( x \approx -2.13 \times 10^{-7} \) for accuracy.

Step 1: Identify the coefficients

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

Step 2: Use the standard quadratic formula

The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute the coefficients into this formula:\[x = \frac{-4,500,000 \pm \sqrt{(4,500,000)^2 - 4 \times 1 \times (-0.96)}}{2 \times 1}\]

Step 3: Simplify inside the square root

Calculate the discriminant \( b^2 - 4ac \):\[(4,500,000)^2 - 4 \times 1 \times (-0.96) = 20,250,000,000,003.84\]

Step 4: Calculate square root and roots

Find \( \sqrt{20,250,000,000,003.84} \) and substitute back into the quadratic formula to solve for both roots:\[x = \frac{-4,500,000 \pm \sqrt{20,250,000,000,003.84}}{2}\]This will yield two roots: a large positive value and a very small negative value.

Step 5: Rationalize the numerator

To find the root near zero, use the alternative formula: \[x = \frac{2c}{-b \mp \sqrt{b^2 - 4ac}}\]Substitute \( c = -0.96 \), \( b = 4,500,000 \), and \( \sqrt{b^2 - 4ac} = \sqrt{20,250,000,000,003.84} \) to get:\[x = \frac{2 \times -0.96}{-4,500,000 + \sqrt{20,250,000,000,003.84}}\]

Step 6: Calculate the root using the rationalized formula

Perform the division with the rationalized formula to approximate the root close to zero. The alternative formula provides a more accurate small root approximation compared to the standard quadratic formula.

Key Concepts

Understanding the Quadratic Formula

The quadratic formula is a powerful mathematical tool which helps find the roots, or solutions, of quadratic equations. A quadratic equation is typically written in the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are coefficients, where \(a\) cannot be zero. The formula for finding the roots of this equation is:

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

This formula uses the coefficients of the equation to determine the values of \(x\) that will satisfy the equation. Notice the plus/minus sign (\(\pm\)) in the equation. This indicates that there will be two possible solutions or roots for \(x\).
In many scenarios, the roots calculated using this formula can either be both real, or one of them can be close to zero. Calculating such small roots accurately is important, especially when dealing with quite large values of \(b\), as in our given equation.

Role of the Discriminant in Quadratic Equations

The discriminant is a specific part of the quadratic formula under the square root, \(b^2 - 4ac\). It helps determine the nature of the roots of a quadratic equation. By examining the value of the discriminant, you can tell whether the roots are real and distinct, real and equal, or complex.

• If \(b^2 - 4ac > 0\), the equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), the equation has exactly one real root, also known as a repeated root.
• If \(b^2 - 4ac < 0\), the equation has two complex roots.

In our solution, the calculation of the discriminant is a crucial step. With a large value of \(b\) as seen here, the discriminant is exceedingly large too. This makes obtaining accurate roots using the quadratic formula more challenging. Using a high precision calculator to find the square root of such a large number can result in errors leading to imprecise solutions.

Rationalizing Expressions to Enhance Accuracy

Rationalizing expressions are often used in mathematical calculations to simplify expressions or calculations that involve square roots. In the context of quadratic equations, rationalizing the numerator can lead to more accurate results, especially when solving for roots that are very small or very large.
Here, we altered the quadratic formula from:

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

To a rationalized form:

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

This transformation helps avoid the issue known as "catastrophic cancellation", where subtracting two nearly equal numbers during computation can lead to a significant loss of precision. With the new formula, we use rationalization to achieve a more accurate value, particularly for the root near zero. This is especially important in cases with considerably large \(b\) values relative to \(ac\), like in the example at hand, improving near-zero root approximations. Rationalizing is a valuable technique for mathematicians and engineers who require high precision in their calculations.