Isolating a in a Radical Equation Chapter 0

Problem setup

• Objective: isolate the variable a in an equation that involves a square root (radical).
• The radical shown is a square root, so we need to undo it by squaring both sides of the equation.
• Core principle: inverse operations. What you do to one side, you must do to the other side to preserve equality.
• The equation given (as interpreted from the transcript) is:
  $\sqrt{\dfrac{5a - 2b}{c}} = 8d - e$
• After squaring both sides, the right-hand side becomes squared, i.e., $(8d - e)^2$ on the right, and the left becomes the radicand squared.
• Plan of steps: square both sides, clear denominators, isolate a, and then simplify the resulting expression.

Step-by-step solution

• Step 1: Square both sides of the original equation
  • Left side after squaring: $\left(\sqrt{\dfrac{5a - 2b}{c}}\right)^2 = \dfrac{5a - 2b}{c}$
  • Right side after squaring: $(8d - e)^2$
  • Resulting equation: $\dfrac{5a - 2b}{c} = (8d - e)^2$
• Step 2: Clear the denominator by multiplying both sides by c
  • $5a - 2b = c\,(8d - e)^2$
• Step 3: Add 2b to both sides to move the constant term on the left
  • $5a = c\,(8d - e)^2 + 2b$
• Step 4: Divide both sides by 5 to isolate a
  • $a = \dfrac{c\,(8d - e)^2 + 2b}{5}$
• Step 5: Optional but helpful – expand the square on the right using FOIL
  • Expand inside: $(8d - e)^2 = 64d^2 - 16de + e^2$
  • Substitution gives: $a = \dfrac{c\,(64d^2 - 16de + e^2) + 2b}{5}$
• Step 6: Distribute c inside the parentheses
  • $c\,(64d^2 - 16de + e^2) = 64d^2c - 16de c + e^2 c$
  • Therefore: $a = \dfrac{64d^2c - 16de c + e^2 c + 2b}{5}$
• Step 7: Provide an equivalent compact form
  • Compact form (using the expanded square): $a = \dfrac{(8d - e)^2\,c + 2b}{5}$
  • Expanded form (as shown in Step 6): $a = \dfrac{64d^2c - 16de c + e^2 c + 2b}{5}$
• Step 8: Summary of final answer forms
  • Final answer in compact form: $a = \dfrac{c\,(8d - e)^2 + 2b}{5}$
  • Final answer in expanded form: $a = \dfrac{64d^2c - 16de c + e^2 c + 2b}{5}$

Key concepts and techniques highlighted

• Inverse operations
  • Radical vs exponent: to undo a square root, square both sides.
  • Addition/subtraction: add 2b to both sides to move terms.
  • Multiplication/division: multiply by c to clear the denominator; divide by 5 to isolate a.
• Algebraic manipulation order
  • Start with removing the radical, then clear fractions, then isolate the variable.
• FOIL/Expanding squares
  • Expanding \((8d - e)^2\) using FOIL yields: \((8d - e)^2 = 64d^2 - 16de + e^2\).
  • Distributing the resulting expression with c gives the expanded numerator: \(64d^2c - 16de c + e^2 c\).
• Alternative forms of the final answer
  • Keeping the square inside: \(a = \dfrac{c(8d - e)^2 + 2b}{5}\).
  • Expanding fully: \(a = \dfrac{64d^2c - 16de c + e^2 c + 2b}{5}\).

Practical considerations and quick checks

• Consistency check: If you square both sides, you may introduce extraneous solutions; check any candidate a in the original equation if solving numerically.
• Real-world relevance: This method demonstrates how to undo radicals and isolate variables in equations with multiple terms and fractions, a common task in algebra and calculus problem solving.

Connections to foundational principles

• Reversal of operations is foundational in solving equations: inverse operations (squaring corresponding to the square root, multiplying to clear denominators, adding to offset subtraction).
• The structure of the problem reinforces the idea that solving for a variable often involves isolating that variable step by step while keeping track of what happens to all other terms.
• The final expanded form illustrates how distributive law and FOIL connect compact and expanded representations of expressions.