If the roots of x^2 + px + q = 0 are in the ratio 2:3, prove that 6p^2 = 25q.

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Step-by-Step Solution

Step 1: Define the roots

We are given that the roots of the quadratic equation x^2 + px + q = 0 are in the ratio 2:3$. Therefore, we can represent the roots as 2\alpha and 3\alpha for some non-zero constant \alpha$.

Step 2: Apply Vieta's formulas for sum of roots

For a quadratic equation ax^2 + bx + c = 0$, the sum of the roots is given by -b/a$. In our equation x^2 + px + q = 0$, a=1$, b=p$, and c=q$. So, the sum of the roots 2\alpha + 3\alpha is equal to -p/1 = -p$. This simplifies to 5\alpha = -p$.

Step 3: Apply Vieta's formulas for product of roots

For a quadratic equation ax^2 + bx + c = 0$, the product of the roots is given by c/a$. In our equation x^2 + px + q = 0$, the product of the roots (2\alpha)(3\alpha) is equal to q/1 = q$. This simplifies to 6\alpha^2 = q$.

Step 4: Express $\alpha$ in terms of $p$

From equation (1), we have 5\alpha = -p$. We can express \alpha in terms of p by dividing both sides by 5, which gives \alpha = -p/5$.

Step 5: Substitute $\alpha$ into the product of roots equation

Now we substitute the expression for \alpha from the previous step into equation (2). This means replacing \alpha with -p/5 in the equation 6\alpha^2 = q$. Squaring -p/5 gives p^2/25$. Multiplying by 6, we get 6p^2/25 = q$.

Step 6: Rearrange to prove the identity

To obtain the desired identity, we multiply both sides of the equation 6p^2/25 = q by 25. This results in 6p^2 = 25q$, which is what we needed to prove.