Question

Solve the equation using square roots. \(2(x-3)^2=24\)

Short answer

The solutions to the equation are \(x = 3 + \sqrt{12}\) and \(x = 3 - \sqrt{12}\)

Step-by-step solution

Simplifying the Equation

Divide both sides of the equation \(2(x-3)^2=24\) by 2. This will give us \( (x-3)^2 = 12 \)

Extracting the Square Root

The next step is to extract the square root from both sides, which results in \(x - 3 = \pm \sqrt{12}\)

Solving for x

Finally, add 3 to both sides to isolate x. This gives \(x = 3 \pm \sqrt{12}\).

Key concepts

Square Roots Method

The square roots method is a straightforward technique for solving quadratic equations that can be written in the form of ax^2 = c. When an equation is presented in this simplified form, we can solve for x by isolating the squared term and taking the square root of both sides. However, an important aspect to remember is to consider both the positive and negative square roots, as squaring a negative number also results in a positive value.

For example, if we have an equation like x^2 = 9, taking the square root of both sides gives us x = \( \pm \sqrt{9} \), or x = \( \pm 3 \). This means that x can be either 3 or -3. Using the square roots method is especially useful when the quadratic equation is already simplified and the squared term is isolated, which makes the solving process more direct and efficient.

Simplify Equations

Simplifying equations is a critical skill that helps make complex problems more manageable. The main goal is to reduce the equation to its most basic form without changing its solution. This involves combining like terms, factoring, dividing by coefficients, and eliminating parentheses. Simplification makes it easier to see the underlying structure of the problem and to apply appropriate methods to find the solution.

In our initial step, we simplified the equation 2(x-3)^2 = 24 by dividing both sides by 2, resulting in (x-3)^2 = 12. This crucial step brought us closer to a form where the square roots method could be applied effectively. Simplifying equations not only helps in solving them but also in checking our work, as a properly simplified equation should yield the same solution when reversed.

Extracting Square Roots

Extracting square roots is a technique used in algebra to solve for a variable within a square. When you extract the square root of both sides of an equation, you’re finding the value that, when squared, gives the original number. It's important to remember that every positive number has two square roots: a positive and a negative root. This is captured by the symbol \( \pm \), indicating both possibilities.

Let's take the equation from our example: (x-3)^2 = 12. By extracting the square root, we get x-3 = \( \pm \sqrt{12} \). However, \sqrt{12} can be further simplified to 2\sqrt{3} because 12 is equal to 4\times3, and square root of 4 is 2. So the final solution can be written as x = 3 \( \pm 2\sqrt{3} \). Recognizing opportunities to simplify the square root, as in this case, can make the answers easier to understand and work with in subsequent calculations.