Without Using A Calculator, Find The Values Of The Integers \[$ A \$\] And \[$ B \$\] For Which The Solution Of The Equation \[$ X \sqrt{24} = X \sqrt{3} + \sqrt{6} \$\] Is \[$ \frac{a + \sqrt{b}}{7} \$\].
Introduction
In this article, we will delve into solving an equation that involves square roots and integers. The equation is given as { x \sqrt{24} = x \sqrt{3} + \sqrt{6} $}$, and we are tasked with finding the values of the integers { a $}$ and { b $}$ for which the solution of the equation is { \frac{a + \sqrt{b}}{7} $}$. We will use algebraic manipulation and properties of square roots to solve for the values of { a $}$ and { b $}$.
Step 1: Simplify the Equation
The first step in solving the equation is to simplify the left-hand side by factoring out the square root of 24. We can write { \sqrt{24} $}$ as { \sqrt{4 \cdot 6} $}$, which simplifies to { 2 \sqrt{6} $}$. Substituting this back into the original equation, we get:
{ x \cdot 2 \sqrt{6} = x \sqrt{3} + \sqrt{6} $}$
Step 2: Isolate the Square Root Terms
Next, we want to isolate the square root terms on one side of the equation. We can do this by subtracting { x \sqrt{3} $}$ from both sides of the equation:
{ 2x \sqrt{6} - x \sqrt{3} = \sqrt{6} $}$
Step 3: Factor Out the Common Term
Now, we can factor out the common term { x $}$ from the left-hand side of the equation:
{ x (2 \sqrt{6} - \sqrt{3}) = \sqrt{6} $}$
Step 4: Solve for x
To solve for { x $}$, we can divide both sides of the equation by { 2 \sqrt{6} - \sqrt{3} $}$:
{ x = \frac{\sqrt{6}}{2 \sqrt{6} - \sqrt{3}} $}$
Step 5: Rationalize the Denominator
To rationalize the denominator, we can multiply both the numerator and denominator by the conjugate of the denominator:
{ x = \frac{\sqrt{6}}{2 \sqrt{6} - \sqrt{3}} \cdot \frac{2 \sqrt{6} + \sqrt{3}}{2 \sqrt{6} + \sqrt{3}} $}$
Simplifying the expression, we get:
{ x = \frac{2 \sqrt{6}^2 + \sqrt{6} \sqrt{3}}{4 \cdot 6 - 3} $}$
{ x = \frac{12 + 3 \sqrt{2}}{21} $}$
Step 6: Find the Values of a and b
Now that we have the value of { x $}$, we can find the values of { a $}$ and { b $}$ by comparing the expression { \frac{a + \sqrt{b}}{7} $}$ with the value of { x $}$:
{ \frac{a + \sqrt{b}}{7} = \frac{12 + 3 \sqrt{2}}{21} $}$
Multiplying both sides of the equation by 7, we get:
{ a + \sqrt{b} = \frac{84 + 21 \sqrt{2}}{21} $}$
Simplifying the expression, we get:
{ a + \sqrt{b} = 4 + \sqrt{2} $}$
Comparing the two expressions, we can see that:
{ a = 4 $}$
{ b = 2 $}$
Conclusion
In this article, we solved the equation { x \sqrt{24} = x \sqrt{3} + \sqrt{6} $}$ without using a calculator. We used algebraic manipulation and properties of square roots to find the values of the integers { a $}$ and { b $}$ for which the solution of the equation is { \frac{a + \sqrt{b}}{7} $}$. We found that { a = 4 $}$ and { b = 2 $}$.
References
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "College Algebra" by James Stewart
Note
Q: What is the main goal of the article?
A: The main goal of the article is to solve the equation { x \sqrt{24} = x \sqrt{3} + \sqrt{6} $}$ without using a calculator and find the values of the integers { a $}$ and { b $}$ for which the solution of the equation is { \frac{a + \sqrt{b}}{7} $}$.
Q: What are the steps involved in solving the equation?
A: The steps involved in solving the equation are:
- Simplify the left-hand side by factoring out the square root of 24.
- Isolate the square root terms on one side of the equation.
- Factor out the common term from the left-hand side of the equation.
- Solve for x.
- Rationalize the denominator.
- Find the values of a and b.
Q: What is the value of x in terms of a and b?
A: The value of x is given by { x = \frac{a + \sqrt{b}}{7} $}$.
Q: How do we find the values of a and b?
A: We find the values of a and b by comparing the expression { \frac{a + \sqrt{b}}{7} $}$ with the value of x.
Q: What are the values of a and b?
A: The values of a and b are { a = 4 $}$ and { b = 2 $}$.
Q: Why is it important to rationalize the denominator?
A: Rationalizing the denominator is important because it allows us to simplify the expression and make it easier to compare with the value of x.
Q: Can we use a calculator to solve the equation?
A: No, we cannot use a calculator to solve the equation. The goal of the article is to solve the equation without using a calculator.
Q: What are the references used in the article?
A: The references used in the article are:
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "College Algebra" by James Stewart
Q: What is the note at the end of the article?
A: The note at the end of the article states that the values of a and b found in the article are specific to the given equation and may not be applicable to other equations.
Q: What is the conclusion of the article?
A: The conclusion of the article is that we have successfully solved the equation { x \sqrt{24} = x \sqrt{3} + \sqrt{6} $}$ without using a calculator and found the values of the integers { a $}$ and { b $}$ for which the solution of the equation is { \frac{a + \sqrt{b}}{7} $}$.