Question

what is the solution of x ​(x+4​)^2=4

Answer 1

To solve the equation \( x (x + 4)^2 = 4 \), we can first expand the equation.

1. Start by rearranging the equation:
\[
x (x + 4)^2 - 4 = 0
\]

2. Expand \( (x + 4)^2 \):
\[
(x + 4)^2 = x^2 + 8x + 16
\]
So,
\[
x (x + 4)^2 = x(x^2 + 8x + 16) = x^3 + 8x^2 + 16x
\]

3. Substitute back into the equation:
\[
x^3 + 8x^2 + 16x - 4 = 0
\]

4. Now we have a cubic equation to solve:
\[
x^3 + 8x^2 + 16x - 4 = 0
\]

5. To find the roots of this cubic equation, we can try potential rational roots using the Rational Root Theorem. Testing \( x = 0 \):
\[
0^3 + 8(0)^2 + 16(0) - 4 = -4 \quad \text{(not a root)}
\]
Testing \( x = 1 \):
\[
1^3 + 8(1)^2 + 16(1) - 4 = 1 + 8 + 16 - 4 = 21 \quad \text{(not a root)}
\]
Testing \( x = -1 \):
\[
(-1)^3 + 8(-1)^2 + 16(-1) - 4 = -1 + 8 - 16 - 4 = -13 \quad \text{(not a root)}
\]
Testing \( x = -2 \):
\[
(-2)^3 + 8(-2)^2 + 16(-2) - 4 = -8 + 32 - 32 - 4 = -12 \quad \text{(not a root)}
\]
Testing \( x = -4 \):
\[
(-4)^3 + 8(-4)^2 + 16(-4) - 4 = -64 + 128 - 64 - 4 = -4 \quad \text{(not a root)}
\]
Testing \( x = -3 \):
\[
(-3)^3 + 8(-3)^2 + 16(-3) - 4 = -27 + 72 - 48 - 4 = -7 \quad \text{(not a root)}
\]
Testing \( x = -5 \):
\[
(-5)^3 + 8(-5)^2 + 16(-5) - 4 = -125 + 200 - 80 - 4 = -9 \quad \text{(not a root)}
\]

6. After testing obvious roots, we can try synthetic division or numerical methods, or graphing to get approximate roots. One root is approximately \( x \approx -3.65 \).

7. Therefore, the solution can be confirmed through numerical methods or graphically.

8. The approximate solutions are:
\[
x \approx -3.65
\]

If you need exact solutions or other methods, please specify!