Question - Finding Equation with Given Roots
Solution:
The question states that $$\alpha$$ and $$\beta$$ are the roots of the equation $$2x^2 - 5x + 3 = 0$$, and it asks us to form an equation whose roots are $$\alpha$$ and $$\beta$$/$$\alpha - \beta$$.First, we need to find the roots $$\alpha$$ and $$\beta$$ by solving the quadratic equation $$2x^2 - 5x + 3 = 0$$.We can use the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, $$a = 2$$, $$b = -5$$, and $$c = 3$$.So, solving for x, we get the roots $$\alpha$$ and $$\beta$$.Next, the roots of the new equation are $$\alpha$$ and $$\beta$$/($$\alpha - \beta$$).The sum of the roots $$S$$ is:\[S = \alpha + \frac{\beta}{\alpha - \beta}\]And the product of the roots $$P$$ is:\[P = \alpha \cdot \frac{\beta}{\alpha - \beta}\]Finally, the equation with these new roots can be written as:\[x^2 - Sx + P = 0\]However, there's a simpler approach without computing the exact roots. We can use transformation of roots directly:If $$\alpha$$ and $$\beta$$ are roots of $$2x^2 - 5x + 3 = 0$$, then the sum of the roots $$\alpha + \beta$$ would be $$5/2$$, and the product $$\alpha \cdot \beta$$ would be $$3/2$$ (according to the properties of quadratic equations, where the sum of the roots is $$-b/a$$ and the product is $$c/a$$).Now for the new roots:The sum of the new roots is $$\alpha + \beta/(\alpha - \beta)$$. We already know that $$\alpha + \beta = 5/2$$. We can simplify $$\beta/(\alpha - \beta)$$ as $$1 - \alpha/(\alpha - \beta)$$. Since we don't know $$\alpha$$ and $$\beta$$ individually, we cannot compute this exactly without assuming values from solving the quadratic, but we can establish a relationship for the equation coefficients based on these expressions.The product of the roots of the new equation is $$\alpha \cdot (\beta/(\alpha - \beta))$$. This simplifies to $$\beta/(\alpha - \beta)$$, which is the same term we have above, and again we run into the problem of not having the specific values of $$\alpha$$ and $$\beta$$.Thus, to solve for the general form of the equation, we would need the specific values of $$\alpha$$ and $$\beta$$, which can be found by solving the given quadratic equation. Once $$\alpha$$ and $$\beta$$ have been determined, the sum and product of the new roots (as described) can be used to form the new quadratic equation.
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