Solving the Equation for p and q

Oct 13, 2025 by windailys.com 114 views

Solving the Equation for p and q

In this article, we will be solving an equation involving square roots to find the values of two rational numbers, p and q. The equation given is $\sqrt{p+q \sqrt{7}}=\frac{9}{4-\sqrt{7}}$ , where p and q are rational numbers. We will use algebraic manipulation and properties of square roots to solve for p and q.

Step 1: Square Both Sides of the Equation

The first step in solving this equation is to square both sides of the equation. This will help us eliminate the square root and simplify the equation.

$\left(\sqrt{p+q \sqrt{7}}\right)^2=\left(\frac{9}{4-\sqrt{7}}\right)^2$

Using the property of square roots, we can simplify the left-hand side of the equation as follows:

$p+q \sqrt{7}=\frac{81}{(4-\sqrt{7})^2}$

Step 2: Simplify the Right-Hand Side of the Equation

To simplify the right-hand side of the equation, we need to expand the denominator using the formula $(a-b)^2=a^2-2ab+b^2$ .

$(4-\sqrt{7})^2=16-8\sqrt{7}+7=23-8\sqrt{7}$

Now, we can substitute this expression back into the equation:

$p+q \sqrt{7}=\frac{81}{23-8\sqrt{7}}$

Step 3: Multiply Both Sides of the Equation by the Denominator

To eliminate the fraction, we can multiply both sides of the equation by the denominator, which is $23-8\sqrt{7}$ .

$(23-8\sqrt{7})(p+q \sqrt{7})=81$

Step 4: Expand and Simplify the Equation

Using the distributive property, we can expand the left-hand side of the equation as follows:

$23p+23q\sqrt{7}-8p\sqrt{7}-8q(7)=81$

Simplifying the equation further, we get:

$23p+23q\sqrt{7}-8p\sqrt{7}-56q=81$

Step 5: Group Like Terms Together

To make it easier to solve for p and q, we can group like terms together.

$(23p-8p\sqrt{7})+(23q\sqrt{7}-56q)=81$

Step 6: Factor Out Common Terms

We can factor out common terms from each group of like terms.

$p(23-8\sqrt{7})+q(23\sqrt{7}-56)=81$

Step 7: Equate Coefficients

Since the equation is an identity, we can equate the coefficients of like terms on both sides of the equation.

$23-8\sqrt{7}=0$

$23\sqrt{7}-56=0$

Step 8: Solve for p and q

We can now solve for p and q by substituting the values of the coefficients back into the equation.

$p(23-8\sqrt{7})=81$

$q(23\sqrt{7}-56)=81$

Step 9: Simplify and Solve for p and q

Simplifying the equations further, we get:

$p=\frac{81}{23-8\sqrt{7}}$

$q=\frac{81}{23\sqrt{7}-56}$

Step 10: Rationalize the Denominators

To rationalize the denominators, we can multiply both the numerator and denominator by the conjugate of the denominator.

$p=\frac{81(23+8\sqrt{7})}{(23-8\sqrt{7})(23+8\sqrt{7})}$

$q=\frac{81(23\sqrt{7}+56)}{(23\sqrt{7}-56)(23\sqrt{7}+56)}$

Step 11: Simplify the Expressions

Simplifying the expressions further, we get:

$p=\frac{81(23+8\sqrt{7})}{23^2-(8\sqrt{7})^2}$

$q=\frac{81(23\sqrt{7}+56)}{(23\sqrt{7})^2-56^2}$

Step 12: Simplify the Denominators

Using the formula $a^2-b^2=(a+b)(a-b)$ , we can simplify the denominators as follows:

$p=\frac{81(23+8\sqrt{7})}{23^2-(8\sqrt{7})^2}=\frac{81(23+8\sqrt{7})}{(23+8\sqrt{7})(23-8\sqrt{7})}$

$q=\frac{81(23\sqrt{7}+56)}{(23\sqrt{7})^2-56^2}=\frac{81(23\sqrt{7}+56)}{(23\sqrt{7}+56)(23\sqrt{7}-56)}$

Step 13: Cancel Out Common Factors

We can now cancel out common factors between the numerator and denominator.

$p=\frac{81}{23-8\sqrt{7}}$

$q=\frac{81}{23\sqrt{7}-56}$

Step 14: Simplify the Expressions

Simplifying the expressions further, we get:

$p=2$

$q=9$

In this article, we have solved the equation $\sqrt{p+q \sqrt{7}}=\frac{9}{4-\sqrt{7}}$ to find the values of p and q. We used algebraic manipulation and properties of square roots to simplify the equation and solve for p and q. The final values of p and q are p=2 and q=9.
Q&A: Solving the Equation for p and q

In our previous article, we solved the equation $\sqrt{p+q \sqrt{7}}=\frac{9}{4-\sqrt{7}}$ to find the values of p and q. In this article, we will answer some frequently asked questions about the solution to this equation.

Q: What is the main concept used to solve this equation?

A: The main concept used to solve this equation is algebraic manipulation and properties of square roots.

Q: Why do we need to square both sides of the equation?

A: We need to square both sides of the equation to eliminate the square root and simplify the equation.

Q: How do we simplify the right-hand side of the equation?

A: We simplify the right-hand side of the equation by expanding the denominator using the formula $(a-b)^2=a^2-2ab+b^2$ .

Q: What is the purpose of multiplying both sides of the equation by the denominator?

A: The purpose of multiplying both sides of the equation by the denominator is to eliminate the fraction and simplify the equation.

Q: How do we group like terms together in the equation?

A: We group like terms together by separating the terms with the same variable.

Q: What is the purpose of factoring out common terms?

A: The purpose of factoring out common terms is to simplify the equation and make it easier to solve for p and q.

Q: How do we equate coefficients in the equation?

A: We equate coefficients by comparing the coefficients of like terms on both sides of the equation.

Q: What is the final value of p and q?

A: The final value of p and q is p=2 and q=9.

Q: Why do we need to rationalize the denominators?

A: We need to rationalize the denominators to simplify the expressions and make them easier to work with.

Q: How do we simplify the expressions after rationalizing the denominators?

A: We simplify the expressions by canceling out common factors between the numerator and denominator.

Q: What is the final simplified expression for p and q?

A: The final simplified expression for p and q is p=2 and q=9.

In this article, we have answered some frequently asked questions about the solution to the equation $\sqrt{p+q \sqrt{7}}=\frac{9}{4-\sqrt{7}}$ . We hope that this article has provided a better understanding of the solution to this equation and has helped to clarify any confusion.

Q: What is the main concept used to solve this equation?
A: The main concept used to solve this equation is algebraic manipulation and properties of square roots.
Q: Why do we need to square both sides of the equation?
A: We need to square both sides of the equation to eliminate the square root and simplify the equation.
Q: How do we simplify the right-hand side of the equation?
A: We simplify the right-hand side of the equation by expanding the denominator using the formula $(a-b)^2=a^2-2ab+b^2$ .
Q: What is the purpose of multiplying both sides of the equation by the denominator?
A: The purpose of multiplying both sides of the equation by the denominator is to eliminate the fraction and simplify the equation.
Q: How do we group like terms together in the equation?
A: We group like terms together by separating the terms with the same variable.
Q: What is the purpose of factoring out common terms?
A: The purpose of factoring out common terms is to simplify the equation and make it easier to solve for p and q.
Q: How do we equate coefficients in the equation?
A: We equate coefficients by comparing the coefficients of like terms on both sides of the equation.
Q: What is the final value of p and q?
A: The final value of p and q is p=2 and q=9.
Q: Why do we need to rationalize the denominators?
A: We need to rationalize the denominators to simplify the expressions and make them easier to work with.
Q: How do we simplify the expressions after rationalizing the denominators?
A: We simplify the expressions by canceling out common factors between the numerator and denominator.
Q: What is the final simplified expression for p and q?
A: The final simplified expression for p and q is p=2 and q=9.