If 3x2
+ 3y2
= 10xy, what is the ratio of x to y?

To solve for the ratio of \(x\) to \(y\) in the equation \(3x^2 + 3y^2 = 10xy\), we can follow these steps:

1. **Simplify the equation:**
   \[ 3x^2 + 3y^2 = 10xy \]
   Divide both sides by 3:
   \[ x^2 + y^2 = \frac{10}{3}xy \]

2. **Rewrite the equation:**
   \[ x^2 + y^2 = \frac{10}{3}xy \]
   Rearrange it to:
   \[ x^2 - \frac{10}{3}xy + y^2 = 0 \]

3. **Solve the quadratic equation:**
   This is a quadratic equation in terms of \(x\). We can use the quadratic formula:
   \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
   Here, \(a = 1\), \(b = -\frac{10}{3}y\), and \(c = y^2\). Plugging these into the quadratic formula:
   \[ x = \frac{\frac{10}{3}y \pm \sqrt{\left(\frac{10}{3}y\right)^2 - 4 \cdot 1 \cdot y^2}}{2 \cdot 1} \]
   \[ x = \frac{\frac{10}{3}y \pm \sqrt{\frac{100}{9}y^2 - 4y^2}}{2} \]
   \[ x = \frac{\frac{10}{3}y \pm \sqrt{\frac{100}{9}y^2 - \frac{36}{9}y^2}}{2} \]
   \[ x = \frac{\frac{10}{3}y \pm \sqrt{\frac{64}{9}y^2}}{2} \]
   \[ x = \frac{\frac{10}{3}y \pm \frac{8}{3}y}{2} \]
   \[ x = \frac{10y \pm 8y}{6} \]

4. **Find the roots:**
   \[ x = \frac{

Learn how to find the ratio of x to y in the equation 3x^2 + 3y^2 = 10xy with a detailed step-by-step solution. This problem involves algebraic manipulation and solving quadratic equations. Suitable for high school students.

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