Solving 8x2 - 2x 0: A Comprehensive Guide to Factoring

Factoring is a powerful method for solving quadratic equations. In this guide, we will walk through the detailed steps to solve the equation 8x2 - 2x 0 by factoring. By the end of this article, you will have a clear understanding of how to approach similar problems.

Factor Out the Common Term

The first step in factoring the equation 8x2 - 2x 0 is to identify and factor out the greatest common factor (GCF) from each term. Here, the GCF of 8x2 and 2x is 2x.

Identify the GCF: 2x Factor out the GCF from each term:

8x2 - 2x 2x(4x - 1) 0

Set Each Factor to Zero

Once the equation is factored, we set each factor equal to zero to solve for x. This is based on the zero product property, which states that if a product of factors equals zero, at least one of the factors must be zero.

Set the first factor equal to zero: Set the second factor equal to zero:

2x 0 or 4x - 1 0

Solve Each Equation

Now, solve each equation individually:

Solving 2x 0: Solving 4x - 1 0:

2x 0

X 0

4x - 1 0

4x 1

x 1/4 or -1/4

Conclusion

Combining the solutions, we find that the solutions to the equation 8x2 - 2x 0 are:

x 0 or x -1/4

Additional Methods for Solving Quadratic Equations

While factoring is a straightforward method, there are other techniques you can use to solve quadratic equations, such as completing the square or the quadratic formula:

Completing the Square

Take half of the x-coefficient and square it: Add and subtract this square inside the equation: Simplify and solve:

8x2 - 2x 1/4 - 1/4 0

(2x - 1/2)2 - 1/4 0

(2x - 1/2)2 1/4

2x - 1/2 ±√(1/4)

2x 1/2 ± 1/2

x 1/4 or -1/4

The Quadratic Formula

For a general quadratic equation ax2 bx c 0, the solutions are given by:

x (-b ± √(b2 - 4ac)) / 2a

In our case, a 8, b -2, and c 0:

x (-(-2) ± √((-2)2 - 4(8)(0))) / 2(8)

x (2 ± √4) / 16

x (2 ± 2) / 16

x 1/4 or -1/4

Conclusion

Understanding how to solve quadratic equations like 8x2 - 2x 0 through factoring, completing the square, and using the quadratic formula is crucial for advanced algebraic and mathematical problem-solving. Each method offers a different perspective and can be useful depending on the specific equation and context.