Question

the real solutions by factoring.
$x^3 - 49x = 0$
what is the solution set? select the correct choice below and fill in any answer boxes within your choice.
a. \\{ \\} (use a comma to separate answers as needed. type an integer or a simplified fraction.)
b. there are no solutions.

Explanation:

Step1: Factor out the GCF

First, factor out the greatest common factor (GCF) from the equation \(x^3 - 49x = 0\). The GCF of \(x^3\) and \(49x\) is \(x\), so we get:
\(x(x^2 - 49) = 0\)

Step2: Factor the difference of squares

Notice that \(x^2 - 49\) is a difference of squares, which can be factored as \((x - 7)(x + 7)\) (since \(a^2 - b^2=(a - b)(a + b)\) with \(a = x\) and \(b = 7\)). So the equation becomes:
\(x(x - 7)(x + 7) = 0\)

Step3: Apply the zero - product property

The zero - product property states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\) (or both). So we set each factor equal to zero:

• For \(x=0\), we have one solution \(x = 0\).
• For \(x - 7=0\), we solve for \(x\) and get \(x=7\).
• For \(x + 7=0\), we solve for \(x\) and get \(x=-7\).

Answer:

\(\{-7, 0, 7\}\)