Solve each problem. The supply and demand equations for a certain commodity are given. supply: p = √(0.1q + 9) - 2 and demand: p = √(25 - 0.1q).
Find the equilibrium demand.

Verified step by step guidance

1. Understand that the equilibrium demand occurs where the supply price equals the demand price, so set the supply equation equal to the demand equation: \(\sqrt{0.1q + 9} - 2 = \sqrt{25 - 0.1q}\).
2. Isolate one of the square root expressions to prepare for squaring both sides. For example, add 2 to both sides to get \(\sqrt{0.1q + 9} = \sqrt{25 - 0.1q} + 2\).
3. Square both sides of the equation to eliminate the square roots. Remember to apply the formula \((a + b)^2 = a^2 + 2ab + b^2\) on the right side.
4. After squaring, simplify the resulting equation by expanding and combining like terms. This will give you a quadratic equation in terms of \(q\).
5. Solve the quadratic equation for \(q\) using factoring, completing the square, or the quadratic formula. Then, check your solutions by substituting back into the original equations to ensure they satisfy the equilibrium condition.

Key Concepts

Equilibrium in Supply and Demand

Equilibrium occurs where the quantity supplied equals the quantity demanded, meaning the supply and demand equations have the same price (p) for a certain quantity (q). Finding equilibrium involves setting the supply equation equal to the demand equation and solving for q.

Solving Equations Involving Square Roots

Both supply and demand equations contain square root expressions. To solve for q, you often need to isolate the square roots and then square both sides carefully to eliminate them, ensuring to check for extraneous solutions introduced by squaring.

Algebraic Manipulation and Equation Solving

After equating the supply and demand functions, algebraic skills are essential to rearrange terms, simplify expressions, and solve for the variable q. This includes combining like terms, isolating variables, and verifying solutions within the domain constraints.