Solve for x: 28x⁸ - 7x⁷ = 0 Using Common Factoring

To solve this problem, we need to apply the following steps:

• Step 1: Identify the highest factor common to all terms and factor it out.
• Step 2: Set each factor equal to zero and solve for $x$.
• Step 3: Validate solutions within the context of the problem statement.

Now, following these steps:

Step 1: Identify and factor out the greatest common factor:

The given equation is $28x^8 - 7x^7 = 0$.

The greatest common factor (GCF) of the terms $28x^8$ and $7x^7$ is $7x^7$.

We can factor the equation as:

$7x^7(4x - 1) = 0$.

Step 2: Set each factor equal to zero:

For $7x^7 = 0$, dividing both sides by 7 yields $x^7 = 0$, which implies $x = 0$.

For $4x - 1 = 0$, solve for $x$:

$4x = 1$

$x = \frac{1}{4}$

Step 3: Verify solutions:

The values $x = 0$ and $x = \frac{1}{4}$ both satisfy the original equation, as substituting them back results in $0$.

Thus, the solutions to the equation are $x = 0$ and $x = \frac{1}{4}$.

The answer, based on the choices provided, is: Answers a and b are correct.