Problem 22 - Algebra and Trigonometry

Question

Solve equation by the square root property. $$ (x-3)^{2}=36 $$

Step-by-Step Solution

Answer

The solutions to the equation are x = 9 and x = -3.

Step 1: Isolate the square

To start with, the square (x-3)^2 needs to be isolated. This is already done in the provided equation (x-3)^2 = 36.

Step 2: Apply square root property

Applying the square root property to both sides of the equation gives two equations: \(x-3 = +\sqrt{36}\) and \(x-3 = -\sqrt{36}\). The square root of 36 is 6, so the equations become \(x - 3 = +6\) and \(x - 3 = -6\).

Step 3: Solve for x

These result in two different values for x after rearranging. From the first equation \(x - 3 = +6\), add 3 to both sides to get \(x = 6 + 3 = 9\). From the second equation \(x - 3 = -6\), add 3 to both sides to get \(x = -6 + 3 = -3\).

Key Concepts

Solving Quadratic Equations

Understanding how to solve quadratic equations is fundamental in algebra. Quadratic equations are in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). To solve them, we use methods like factoring, completing the square, the quadratic formula, or the square root property.

The method you choose often depends on the equation's form. When the equation is a perfect square trinomial, like \( (x-3)^2 = 36 \), the square root property is an efficient way to find \(x\). After taking square roots, remember that you obtain two solutions because squaring either a positive or negative value gives you the same result - in this case, 36.

Isolate the Square

The first step in solving a quadratic equation by the square root method is to isolate the squared term. This means that you want the term which is squared (in this case, \( (x-3)^2 \) ) to be by itself on one side of the equation.

Isolating the square makes applying the square root property simpler as it gets rid of the square and allows you to deal with what's inside of it directly. In our exercise, \( (x-3)^2 = 36 \), the square is already isolated. If there had been additional terms or a coefficient in front of the squared term, you would need to move or divide these first.

Applying Square Roots

Once the square is isolated, you apply the square root property. This step involves taking the square root of both sides of the equation, which effectively undoes the square and brings you closer to the solution.

However, it is vital to consider both the positive and negative square roots. Mathematically, this is symbolized as \( \pm \sqrt{n} \) for any non-negative number \( n \). It leads to two separate linear equations. In the example of \( (x-3)^2 = 36 \), applying square roots gives you \( x-3 = \pm 6 \) since \( \sqrt{36} = 6 \). These two possibilities need to be solved separately to find the two potential values for \( x \).

Remember that applying the square root property is valid only when the equation has been correctly simplified and the square isolated. This method emphasizes the importance of checking your work to ensure that both potential solutions are valid in the original equation's context.