Solving the Quadratic Equation: 3x2 27

The quadratic equation has a wide range of applications in mathematics, physics, and engineering. One such example is the equation 3x2 27. This article will guide you through the process of transforming this equation into its standard form and solving it step by step.

Transforming the Equation

Let's start with the given equation: 3x2 27.

Move all terms to one side of the equation:

Divide the entire equation by 3 to simplify:

Now, we have the quadratic equation in its standard form: x2 - 9 0.

Factoring and Solving the Quadratic Equation

The next step is to factorize the equation. This can be achieved as follows:

(x - 3)(x 3) 0

By factoring the equation, we have identified two potential solutions for x: x - 3 0 which implies x 3 x 3 0 which implies x -3

Therefore, the solutions to the equation 3x2 27 are:

x 3 and x -3

Alternative Method: Square Root Property

There is another method to solve this equation: the square root property. Let's revisit the simplified equation:

x2 9

According to the square root property, if x2 9, then:

x plusmn;3

This means that x can be either 3 or -3. Therefore, the solutions to the equation are the same as found through factoring.

Conclusion

The equation 3x2 27 is transformed into its standard form as x2 - 9 0, which can be factored to (x - 3)(x 3) 0, resulting in solutions x 3 and x -3. Alternatively, you can use the square root property to find these solutions by recognizing that x2 9 has solutions x plusmn;3.

Frequently Asked Questions (FAQs)

Q1: What are the steps to solve a quadratic equation?

A1: To solve a quadratic equation, follow these steps: Move all terms to one side of the equation. Divide the equation by the coefficient of x2. Factorize the equation or use the square root property. Find the solutions using the factored form or square root.

Q2: Can you provide an example of how to apply the square root property?

A2: Sure! Consider the equation x2 9. Using the square root property, we find that x plusmn;3. This means x can be 3 or -3.

Q3: How are quadratic equations used in real life?

A3: Quadratic equations are used in various real-life applications, such as calculating the trajectory of a projectile, optimizing the design of structures, and analyzing financial data. They play a crucial role in many fields including physics, engineering, and economics.