Solve each equation in Exercises 15–34 by the square root property.$(3x+2{)}^2=9$

Verified step by step guidance

1. Identify the equation given: \( (3x + 2)^2 = 9 \). This is a perfect setup for the square root property because the variable expression is squared.
2. Apply the square root property, which states that if \( A^2 = B \), then \( A = \pm \sqrt{B} \). Here, set \( 3x + 2 = \pm \sqrt{9} \).
3. Simplify the square root: \( \sqrt{9} = 3 \), so rewrite the equation as \( 3x + 2 = \pm 3 \). This gives two separate equations to solve: \( 3x + 2 = 3 \) and \( 3x + 2 = -3 \).
4. Solve each linear equation separately. For \( 3x + 2 = 3 \), subtract 2 from both sides to isolate the term with \( x \). For \( 3x + 2 = -3 \), do the same.
5. Finally, divide both sides of each equation by 3 to solve for \( x \). This will give you the two possible solutions for \( x \).

Key Concepts

Square Root Property

The square root property states that if an equation is in the form (expression)^2 = k, then the solution can be found by taking the square root of both sides, resulting in expression = ±√k. This method is useful for solving quadratic equations that are already isolated as a perfect square.

Isolating the Squared Term

Before applying the square root property, it is essential to isolate the squared term on one side of the equation. This means manipulating the equation so that (3x + 2)^2 stands alone, allowing you to take the square root of both sides accurately.

Solving Linear Equations After Taking Square Roots

After applying the square root property, you get two linear equations due to the ± sign. Solving these linear equations involves isolating the variable x by performing inverse operations such as addition, subtraction, multiplication, or division.