Solve the equation.

Question

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
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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).