Example Question - sum of roots
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Sum of Roots of a Quadratic Equation
Для квадратного уравнения вида \( ax^2 + bx + c = 0 \) сумма корней вычисляется по формуле Виета: \( x_1 + x_2 = -\frac{b}{a} \). <p>В данном случае у нас есть уравнение \( 5x^2 - 7x + 1 = 0 \), где \( a = 5 \), \( b = -7 \), и \( c = 1 \).</p> <p>Используем формулу Виета для нахождения суммы корней:</p> <p>\( x_1 + x_2 = -\frac{-7}{5} \)</p> <p>\( x_1 + x_2 = \frac{7}{5} \).</p> <p>Следовательно, сумма корней данного уравнения равна \( \frac{7}{5} \).</p>
Finding Equation with Given Roots
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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