Question
X (x+ 16) squared / 9 = 7
Answers
GPT-4o mini
To solve the equation \(\frac{X (X + 16)^2}{9} = 7\), we can follow these steps:
1. Multiply both sides by 9 to eliminate the denominator:
\[
X (X + 16)^2 = 63
\]
2. Expand the left side:
\[
X (X^2 + 32X + 256) = 63
\]
\[
X^3 + 32X^2 + 256X = 63
\]
3. Move 63 to the left side:
\[
X^3 + 32X^2 + 256X - 63 = 0
\]
Now we need to solve the cubic equation \(X^3 + 32X^2 + 256X - 63 = 0\). This can be done with various methods, including numerical methods or using the Rational Root Theorem to find possible rational roots.
Let's check for possible rational roots. Testing \(X = 1\):
\[
1^3 + 32(1^2) + 256(1) - 63 = 1 + 32 + 256 - 63 = 226 \quad (\text{not a root})
\]
Testing \(X = -1\):
\[
(-1)^3 + 32(-1)^2 + 256(-1) - 63 = -1 + 32 - 256 - 63 = -288 \quad (\text{not a root})
\]
Testing \(X = 3\):
\[
3^3 + 32(3^2) + 256(3) - 63 = 27 + 32(9) + 256(3) - 63
\]
\[
= 27 + 288 + 768 - 63 = 1020 \quad (\text{not a root})
\]
Testing \(X = -3\):
\[
(-3)^3 + 32(-3)^2 + 256(-3) - 63 = -27 + 32(9) - 768 - 63
\]
\[
= -27 + 288 - 768 - 63 = -570 \quad (\text{not a root})
\]
Testing \(X = -2\):
\[
(-2)^3 + 32(-2)^2 + 256(-2) - 63 = -8 + 32(4) - 512 - 63
\]
\[
= -8 + 128 - 512 - 63 = -455 \quad (\text{not a root})
\]
Testing \(X = 2\):
\[
(2)^3 + 32(2^2) + 256(2) - 63 = 8 + 32(4) + 512 - 63
\]
\[
= 8 + 128 + 512 - 63 = 585 \quad (\text{not a root})
\]
Continuing this process, we can use numerical methods or a graphing approach to approximate the roots. Another option is to use synthetic division or the cubic formula, which can also be complicated without the assistance of computational tools for specific results.
If you have a graphing calculator or software, you can also graph the equation \(X^3 + 32X^2 + 256X - 63 = 0\) to find the approximate roots.
Please let me know how you would like to proceed or if you would prefer to find further roots or an alternative approach.
1. Multiply both sides by 9 to eliminate the denominator:
\[
X (X + 16)^2 = 63
\]
2. Expand the left side:
\[
X (X^2 + 32X + 256) = 63
\]
\[
X^3 + 32X^2 + 256X = 63
\]
3. Move 63 to the left side:
\[
X^3 + 32X^2 + 256X - 63 = 0
\]
Now we need to solve the cubic equation \(X^3 + 32X^2 + 256X - 63 = 0\). This can be done with various methods, including numerical methods or using the Rational Root Theorem to find possible rational roots.
Let's check for possible rational roots. Testing \(X = 1\):
\[
1^3 + 32(1^2) + 256(1) - 63 = 1 + 32 + 256 - 63 = 226 \quad (\text{not a root})
\]
Testing \(X = -1\):
\[
(-1)^3 + 32(-1)^2 + 256(-1) - 63 = -1 + 32 - 256 - 63 = -288 \quad (\text{not a root})
\]
Testing \(X = 3\):
\[
3^3 + 32(3^2) + 256(3) - 63 = 27 + 32(9) + 256(3) - 63
\]
\[
= 27 + 288 + 768 - 63 = 1020 \quad (\text{not a root})
\]
Testing \(X = -3\):
\[
(-3)^3 + 32(-3)^2 + 256(-3) - 63 = -27 + 32(9) - 768 - 63
\]
\[
= -27 + 288 - 768 - 63 = -570 \quad (\text{not a root})
\]
Testing \(X = -2\):
\[
(-2)^3 + 32(-2)^2 + 256(-2) - 63 = -8 + 32(4) - 512 - 63
\]
\[
= -8 + 128 - 512 - 63 = -455 \quad (\text{not a root})
\]
Testing \(X = 2\):
\[
(2)^3 + 32(2^2) + 256(2) - 63 = 8 + 32(4) + 512 - 63
\]
\[
= 8 + 128 + 512 - 63 = 585 \quad (\text{not a root})
\]
Continuing this process, we can use numerical methods or a graphing approach to approximate the roots. Another option is to use synthetic division or the cubic formula, which can also be complicated without the assistance of computational tools for specific results.
If you have a graphing calculator or software, you can also graph the equation \(X^3 + 32X^2 + 256X - 63 = 0\) to find the approximate roots.
Please let me know how you would like to proceed or if you would prefer to find further roots or an alternative approach.