Solving Equations Using the Square Root Property

Equations and Inequalities

Solving Equations by the Square Root Property

The square root property is a useful algebraic tool for solving equations of the form $x^2 = a$. This method is commonly used in Precalculus to find solutions to quadratic equations that can be rewritten in squared form.

• Square Root Property: If $x^2 = a$, then $x = \sqrt{a}$ or $x = -\sqrt{a}$.

• Application: This property allows us to solve for $x$ by taking the square root of both sides of the equation, remembering to include both the positive and negative roots.

• Exact Answers: Solutions should be expressed in radical form when possible, unless the radicand is a perfect square.

Example

• Given Equation: $x^2 = 9$

• Solution: Take the square root of both sides:

  • $x = \sqrt{9}$ or $x = -\sqrt{9}$

  • $x = 3$ or $x = -3$

• Solution Set: $\{3, -3\}$

General Steps for Solving by the Square Root Property

1. Isolate the squared term on one side of the equation.

2. Apply the square root property to solve for the variable.

3. Simplify the radical if possible.

4. Express the solution set using exact values (radicals) and separate multiple solutions with commas.

Additional info:

• If the equation is not already in the form $x^2 = a$, rearrange or complete the square as needed before applying the property.

• For equations like $(x - h)^2 = k$, the solutions are $x = h + \sqrt{k}$ and $x = h - \sqrt{k}$.