Question

Solve the equation.
left parenthesis x plus 7 right parenthesis squared left parenthesis x minus 2 right parenthesis left parenthesis 4 x minus 7 right parenthesis equals 0
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Part 1
x​ =
  
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​(Use a comma to separate answers as needed. Type an integer or a simplified​ fraction.)

Answer 1

To solve the equation

\[
(x + 7)^2 (x - 2)(4x - 7) = 0,
\]

we will set each factor equal to zero and solve for \(x\).

1. First, set \((x + 7)^2 = 0\):

\[
(x + 7) = 0 \implies x = -7.
\]

Since it is a squared term, \(x = -7\) is a double root.

2. Next, set \((x - 2) = 0\):

\[
x - 2 = 0 \implies x = 2.
\]

3. Finally, set \((4x - 7) = 0\):

\[
4x - 7 = 0 \implies 4x = 7 \implies x = \frac{7}{4}.
\]

Now, we can summarize the solutions:

- From \((x + 7)^2=0\): \(x = -7\) (double root)
- From \((x - 2)=0\): \(x = 2\)
- From \((4x - 7)=0\): \(x = \frac{7}{4}\)

Thus, the complete set of solutions to the equation is:

\[
x = -7, 2, \frac{7}{4}.
\]

So the final answer is:

\[
-7, 2, \frac{7}{4}
\]

(You can enter this as: -7, 2, 7/4).