Sure! Let's solve the quadratic equation [tex]\(3x^2 - 8xy - 3y^2 = 0\)[/tex] for [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex].
### Step-by-Step Solution:
1. Identify the general form of the quadratic equation. In this case, the equation is:
[tex]\[
3x^2 - 8xy - 3y^2 = 0
\][/tex]
2. Recognize that this is a quadratic equation in [tex]\(x\)[/tex], where the coefficients are functions of [tex]\(y\)[/tex].
3. Write down the quadratic formula, which is used to solve for [tex]\(x\)[/tex] in the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For our given equation [tex]\(3x^2 - 8xy - 3y^2 = 0\)[/tex]:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = -8y\)[/tex]
- [tex]\(c = -3y^2\)[/tex]
4. Substitute the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the quadratic formula:
[tex]\[
x = \frac{-(-8y) \pm \sqrt{(-8y)^2 - 4 \cdot 3 \cdot (-3y^2)}}{2 \cdot 3}
\][/tex]
Simplify the values:
[tex]\[
x = \frac{8y \pm \sqrt{64y^2 + 36y^2}}{6}
\][/tex]
[tex]\[
x = \frac{8y \pm \sqrt{100y^2}}{6}
\][/tex]
[tex]\[
x = \frac{8y \pm 10y}{6}
\][/tex]
5. Solve for the two possible values of [tex]\(x\)[/tex]:
- First solution:
[tex]\[
x = \frac{8y + 10y}{6} = \frac{18y}{6} = 3y
\][/tex]
- Second solution:
[tex]\[
x = \frac{8y - 10y}{6} = \frac{-2y}{6} = - \frac{y}{3}
\][/tex]
### Conclusion:
The solutions for [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex] are:
[tex]\[
x = 3y \quad \text{or} \quad x = -\frac{y}{3}
\][/tex]
So, the solutions to the equation [tex]\(3x^2 - 8xy - 3y^2 = 0\)[/tex] are:
[tex]\[
x = 3y \quad \text{and} \quad x = -\frac{y}{3}
\][/tex]
