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Chapter 10: Problem 78

Determine whether \(3-\sqrt{13}\) is a solution to the equation \(x^{2}-6 x=3\).

Short Answer

Expert verified
No, \(3-\sqrt{13}\) is not a solution to the equation.

Step by step solution

01

- Substitute

Substitute the given value, which is \(x = 3 - \sqrt{13}\), into the equation \(x^2 - 6x = 3\).
02

- Compute \(x^2\)

Compute \(x^2\) for \(x = 3 - \sqrt{13}\): \(x^2 = (3 - \sqrt{13})^2\). Using the binomial theorem: \[(3 - \sqrt{13})^2 = 3^2 - 2\cdot 3\cdot \sqrt{13} + (\sqrt{13})^2\] \[(3 - \sqrt{13})^2 = 9 - 6\sqrt{13} + 13\] \[(3 - \sqrt{13})^2 = 22 - 6\sqrt{13}\]
03

- Compute \(-6x\)

Compute \(-6x\) for \(x = 3 - \sqrt{13}\): \[ -6x = -6(3 - \sqrt{13})\] \[ -6x = -18 + 6\sqrt{13}\]
04

- Add \(x^2\) and \(-6x\)

Now add \(x^2\) and \(-6x\) and check if the result equals 3: \[ 22 - 6\sqrt{13} + (-18 + 6\sqrt{13}) = 22 - 18\] \[ 4 = 3\]
05

- Verify if the result meets the equation

Verify the obtained result 4 against the right side of the equation which is 3. Since 4 does not equal 3, \(3 - \sqrt{13}\) is not a solution of the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method
The substitution method is a powerful way to determine if a given value is a solution to an equation. In this problem, the given value is \( x = 3 - \sqrt{13} \). To use the substitution method, we plug this value into the original equation \( x^2 - 6x = 3 \).

Here are the substitution steps:
  • Replace \( x \) in the equation with \( 3 - \sqrt{13} \).
This helps simplify the equation to see if the left side can equal the right side (3). While it might look challenging, following each step methodically ensures accuracy.
Binomial Theorem
The binomial theorem lets us expand expressions that are raised to a power, and it's especially useful when working with quadratics.

In this problem, we use the binomial theorem to compute \( (3 - \sqrt{13})^2 \):
  • Expand \( (3 - \sqrt{13})^2 \) using the formula: \[ (a - b)^2 = a^2 - 2ab + b^2 \]
So, \[(3 - \sqrt{13})^2 = 3^2 - 2 \cdot 3 \cdot \sqrt{13} + (\sqrt{13})^2 \]Simplifying each term:
  • \( 3^2 = 9 \)
  • \( -2 \cdot 3 \cdot \sqrt{13} = -6\sqrt{13} \)
  • \( (\sqrt{13})^2 = 13 \)
Combining these, we get:
\[ (3 - \sqrt{13})^2 = 9 - 6\sqrt{13} + 13 = 22 - 6\sqrt{13} \]
Verifying Solutions
Verifying a solution involves checking if substituting the value back into the equation results in a true statement. After calculating \( x^2 \) and \( -6x \), we add these to verify:

From step 2, we have:
\[ x^2 = (3 - \sqrt{13})^2 = 22 - 6 \sqrt{13} \]From step 3:
\[ -6x = -6(3 - \sqrt{13}) = -18 + 6 \sqrt{13} \]

Adding these results:
\[ x^2 + (-6x) = 22 - 6 \sqrt{13} + (-18 + 6 \sqrt{13}) = 22 - 18 = 4 \]

Since 4 does not equal 3, we confirm: \( 3 - \sqrt{13} \) is not a solution of the equation.

Always verify your results to avoid errors and confirm the correctness of the solution.

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Most popular questions from this chapter

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