Question

Найдите сумму и произведение корней уравнения 3x^+5x-1=0. СРОЧНООООООООООООО

Answer 1

Посчитаем Дискриминант
Д=(в)в квадрате-4ас
Д=25+12=37
х=(-5+корень из 37)/2
х=(-5-корень из 37)/2

Answer 2

Solving the Equation 3x^2 + 5x - 1 = 0

To find the sum and product of the roots of the quadratic equation 3x^2 + 5x - 1 = 0, we can use the quadratic formula:

Quadratic Formula: The solutions to the quadratic equation ax^2 + bx + c = 0 are given by: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]

Finding the Roots

Using the quadratic formula, we can find the roots of the given equation:

\[ x = \frac{{-5 \pm \sqrt{{5^2 - 4*3*(-1)}}}}{{2*3}} \]

\[ x = \frac{{-5 \pm \sqrt{{25 + 12}}}}{{6}} \]

\[ x = \frac{{-5 \pm \sqrt{{37}}}}{{6}} \]

So, the roots of the equation 3x^2 + 5x - 1 = 0 are: \[ x_1 = \frac{{-5 + \sqrt{{37}}}}{{6}} \] \[ x_2 = \frac{{-5 - \sqrt{{37}}}}{{6}} \]

Sum and Product of the Roots

The sum of the roots (denoted as S) and the product of the roots (denoted as P) can be calculated using the following relationships:

Sum of the Roots: \[ S = x_1 + x_2 \]

Product of the Roots: \[ P = x_1 * x_2 \]

Calculating the Sum and Product

Let's calculate the sum and product of the roots using the values we found:

\[ S = \frac{{-5 + \sqrt{{37}}}}{{6}} + \frac{{-5 - \sqrt{{37}}}}{{6}} \]

\[ S = \frac{{-10}}{{6}} = -\frac{5}{3} \]

\[ P = \left(\frac{{-5 + \sqrt{{37}}}}{{6}}\right) * \left(\frac{{-5 - \sqrt{{37}}}}{{6}}\right) \]

\[ P = \frac{{25 - 37}}{{36}} = -\frac{3}{2} \]

Conclusion

Therefore, the sum of the roots is -5/3 and the product of the roots is -3/2.