Problem 23 - Beginning and Intermediate Algebra

Question

Solve each equation for the specified variable. (Leave \(\pm\) in the answers.) \(p=\frac{E^{2} R}{(r+R)^{2}}\) for \(R \quad(\) where \(E>0)\)

Step-by-Step Solution

Answer

R = \left(\frac{E^{2} - 2pr \pm \sqrt{(E^{2} - 2pr)^{2} - 4p^{2}r^{2}}}{2p} \right)

Step 1: Isolate the equation

Start with the given equation: o\[p = \frac{E^{2} R}{(r+R)^{2}}\]

Step 2: Multiply both sides by \( (r + R)^{2} \)

Multiply both sides of the equation by \( (r + R)^{2} \) to eliminate the denominator on the right side:o\[ p (r + R)^{2} = E^{2} R\]

Step 3: Expand and rearrange the equation

Expand the squared term and rearrange the equation to bring all terms involving R to one side:o\[ p (r^2 + 2rR + R^{2}) = E^{2}R\]

Step 4: Collect all terms involving R on one side

Distribute the \p\ across the terms inside the parentheses and then collect all terms involving \R\ on one side:o\[ pr^2 + 2prR + pR^{2} = E^{2}R\]\[ pr^2 = E^{2}R - 2prR - pR^{2}\]

Step 5: Solve the quadratic equation

Rewrite the equation in standard quadratic form \( AX^2+ BX + C = 0 \) and solve for \R\:o\[pR^{2} - (E^{2} - 2pr)R + pr^{2} = 0\]Using the quadratic formula, \( R = \frac{-B \pm \sqrt{B^{2} - 4AC}}{2A} \), solve for \R\ where \ A=p, B= - (E^{2} - 2pr) \, and \ C=pr^{2} \:o\[ R = \frac{-( - (E^{2} - 2pr)) \pm \sqrt{((E^{2} - 2pr))^{2} - 4p(pr^{2})}}{2p}\]

Step 6: Simplify the solution

Simplify the expression under the square root and finalize the solution:o\[ R = \frac{(E^{2} - 2pr) \pm \sqrt{(E^{2} - 2pr)^{2} - 4(p)(pr^{2})}}{2p}\]

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate or solve for specific variables. This process often includes multiplying, dividing, adding, or subtracting terms on both sides.
It can simplify complicated expressions, making them easier to work with.
To solve equations, follow these steps:

• Identify the terms that need manipulation.
• Apply algebraic operations to both sides of the equation.
• Rearrange the equation to isolate the desired variable.

For example, in our exercise:
Start with the given equation: \[ p = \frac{E^{2} R}{(r+R)^{2}} \] Multiply both sides by \[ (r + R)^{2} \] to remove the denominator:
\[ p (r + R)^{2} = E^{2} R \] This step simplifies the equation, making it easier to isolate and solve for R.

Isolating Variables

Isolating variables means rearranging an equation so that the variable you're solving for is alone on one side of the equation.
This helps in simplifying the problem and finding the solution. Steps include:

• Combine like terms involving the variable.
• Use inverse operations to move other terms to the opposite side.

In our example, after multiplying both sides by \[ (r + R)^{2} \] we have:
\[ p (r^2 + 2rR + R^{2}) = E^{2} R \] Distribute \[ p \] and collect all R terms on one side:
\