Question
I'm having trouble solving for 'y' in the equation x=(y^2+3y)^(1/3). Can you find an explanation or solution to this problem?
Answers
GPT-4o mini
To solve the equation \( x = (y^2 + 3y)^{1/3} \) for \( y \), we will first eliminate the cube root by cubing both sides of the equation. Here's a step-by-step breakdown:
1. Start with the original equation:
\[
x = (y^2 + 3y)^{1/3}
\]
2. Cube both sides to isolate \( y^2 + 3y \):
\[
x^3 = y^2 + 3y
\]
3. Rearranging the equation gives:
\[
y^2 + 3y - x^3 = 0
\]
Now we have a quadratic equation in terms of \( y \). The standard form of a quadratic equation is \( ay^2 + by + c = 0 \). For our equation:
- \( a = 1 \)
- \( b = 3 \)
- \( c = -x^3 \)
4. We can use the quadratic formula to solve for \( y \):
\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
5. Substituting the values of \( a \), \( b \), and \( c \):
\[
y = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-x^3)}}{2 \cdot 1}
\]
\[
y = \frac{-3 \pm \sqrt{9 + 4x^3}}{2}
\]
6. Therefore, the solutions for \( y \) are:
\[
y = \frac{-3 + \sqrt{9 + 4x^3}}{2}
\]
and
\[
y = \frac{-3 - \sqrt{9 + 4x^3}}{2}
\]
These represent two possible values for \( y \) given a specific \( x \). Depending on the context or limitations on \( y \), you might choose one solution over the other.
1. Start with the original equation:
\[
x = (y^2 + 3y)^{1/3}
\]
2. Cube both sides to isolate \( y^2 + 3y \):
\[
x^3 = y^2 + 3y
\]
3. Rearranging the equation gives:
\[
y^2 + 3y - x^3 = 0
\]
Now we have a quadratic equation in terms of \( y \). The standard form of a quadratic equation is \( ay^2 + by + c = 0 \). For our equation:
- \( a = 1 \)
- \( b = 3 \)
- \( c = -x^3 \)
4. We can use the quadratic formula to solve for \( y \):
\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
5. Substituting the values of \( a \), \( b \), and \( c \):
\[
y = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-x^3)}}{2 \cdot 1}
\]
\[
y = \frac{-3 \pm \sqrt{9 + 4x^3}}{2}
\]
6. Therefore, the solutions for \( y \) are:
\[
y = \frac{-3 + \sqrt{9 + 4x^3}}{2}
\]
and
\[
y = \frac{-3 - \sqrt{9 + 4x^3}}{2}
\]
These represent two possible values for \( y \) given a specific \( x \). Depending on the context or limitations on \( y \), you might choose one solution over the other.