How did you know to use the number 636 for your x in the hit and trial method and not use another number?
How did you know to use the number 636 for your x in the hit and trial method and not use another number?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
How did you know to use the number 636 for your x in the hit and trial method and not use another number?
![### Solving a Cubic Equation using the Trial and Error Method
In this example, we will solve the given cubic equation using the trial and error method and the quadratic formula.
#### Step 2(b)
**Given Equation:**
\[ x^3 + 3x^2 + 2x - 258,474,216 = 0 \tag{1} \]
**Method:**
Using the trial and error method, we assume a root:
\[ x_1 = 636 \]
Substitute \( x = 636 \) into the equation:
\[ (636)^3 + 3(636)^2 + 2(636) - 258,474,216 = 0 \]
\[ 0 = 0 \]
This confirms that \( x = 636 \) is a root of equation (1).
**Factoring:**
Since \( x = 636 \) is a root, \( x - 636 \) is a factor.
Divide \( x^3 + 3x^2 + 2x - 258,474,216 \) by \( x - 636 \).
\[
\begin{array}{r|rr}
& x^2 + 639x + 406,406 & \\
x-636 & x^3 + 3x^2 + 2x - 258,474,216 \\
\cline{2-2}
& (x^3 - 636x^2) & \\
& x^3 - 636x^2 & - 258,474,216 \\
& 636x^2 + 2x & \\
& 636x^2 - 406,406x \\
\cline{2-2}
& (2x - 406,406x) & - 258,474,216 \\
& 406,406x - 258,474,216 & \\
\end{array}
\]
This simplifies to the quotient:
\[ x^2 + 639x + 406,406 = 0 \]
**Solving the Quadratic Equation:**
Using the quadratic formula \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \):
For
\[ x^2 + 639x + 406,406 = 0 \]
Where
\[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc965ec1f-6ecd-467d-b5f1-022eb1dfa0a4%2F4bf1e206-de40-4159-9721-9cc0ed92a732%2Fa7vtos_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving a Cubic Equation using the Trial and Error Method
In this example, we will solve the given cubic equation using the trial and error method and the quadratic formula.
#### Step 2(b)
**Given Equation:**
\[ x^3 + 3x^2 + 2x - 258,474,216 = 0 \tag{1} \]
**Method:**
Using the trial and error method, we assume a root:
\[ x_1 = 636 \]
Substitute \( x = 636 \) into the equation:
\[ (636)^3 + 3(636)^2 + 2(636) - 258,474,216 = 0 \]
\[ 0 = 0 \]
This confirms that \( x = 636 \) is a root of equation (1).
**Factoring:**
Since \( x = 636 \) is a root, \( x - 636 \) is a factor.
Divide \( x^3 + 3x^2 + 2x - 258,474,216 \) by \( x - 636 \).
\[
\begin{array}{r|rr}
& x^2 + 639x + 406,406 & \\
x-636 & x^3 + 3x^2 + 2x - 258,474,216 \\
\cline{2-2}
& (x^3 - 636x^2) & \\
& x^3 - 636x^2 & - 258,474,216 \\
& 636x^2 + 2x & \\
& 636x^2 - 406,406x \\
\cline{2-2}
& (2x - 406,406x) & - 258,474,216 \\
& 406,406x - 258,474,216 & \\
\end{array}
\]
This simplifies to the quotient:
\[ x^2 + 639x + 406,406 = 0 \]
**Solving the Quadratic Equation:**
Using the quadratic formula \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \):
For
\[ x^2 + 639x + 406,406 = 0 \]
Where
\[
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