Question

Найдите наименьшее значение суммы квадратов корней уравнения (a - параметр).​

Answer 1

Ответ:

$Min_{(x_1^2+x_2^2)}=8$

Решение

Рассмотрим уравнение
$x^2-2ax+2a^2-6a+8=0$

Согласно теореме Виета сумма корней приведенного квадратного уравнения равна среднему коэффициенту взятого с противоположным знаком, а произведение равно свободному члену:
$x_1+x_2=-b\\x_1*x_2=c$

Найдем отсюда сумму квадратов:
$(x_1+x_2)^2=x_1^2+2x_1x_2+x_2^2\\ x_1^2+x_2^2=(x_1+x_2)^2-2x_1x_2\\x_1^2+x_2^2=b^2-2c$

Пусть $x_1^2+x_2^2=f_{(a)}$, тогда подставим значения коэффициентов и найдем минимальное значение функции:
$f_{(a)}=4a^2-2(2a^2-6a+8)\\f_{(a)}=4a^2-4a^2+12a-16\\f_{(a)}=12a-16$

Функция является линейной, с положительным коэффициентом при a, то есть монотонно возрастает, значит меньшему значению аргумента соответствует меньшее значение функции
Из дискриминанта видно, что область определения a при которых имеются действительные решения равна [2;4]

Следовательно минимальное значение f(a) лежит в меньшем значении промежутка [2;4], то есть в a = 2

$f_{(2)}=12*2-16\\f_{(2)}=8\\Min_{(x_1^2+x_2^2)}=8$

Полные Формулы

$ax^2+bx+c=0$, тогда для действительных корней теорема Виета:

$x_1+x_2=-\frac{b}{a} \\x_1*x_2=\frac{c}{a}$

Формула корней через дискриминант:
$D=b^2-4ac\\x_{1,2}=\frac{-b \pm \sqrt{D} }{2a} \\x_{1,2}=\frac{-b \pm \sqrt{b^2-4ac} }{2a}$

Или половинный дискриминант (если b кратно 2):
$D=(\frac{b}{2}) ^2-ac\\x_{1,2}=\frac{\frac{b}{2} \pm \sqrt{D} }{a} \\x_{1,2}=\frac{\frac{b}{2} \pm \sqrt{(\frac{b}{2}) ^2-ac} }{a}$

Все это при $D\geq 0$

Answer 2

Finding the Minimum Value of the Sum of Squares of the Roots of the Equation

To find the minimum value of the sum of squares of the roots of the equation, we need to determine the nature of the equation and its roots. Unfortunately, the provided search results do not directly address the specific equation mentioned in the question. However, we can still discuss some general concepts related to finding the minimum value of the sum of squares of the roots.

In general, to find the minimum value of the sum of squares of the roots of a quadratic equation, we can use the concept of discriminant. The discriminant of a quadratic equation determines the nature of its roots. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has two identical real roots. And if the discriminant is negative, the equation has two complex roots.

Let's consider a quadratic equation in the form of ax^2 + bx + c = 0, where a, b, and c are constants.

The discriminant (D) of the quadratic equation is given by the formula: D = b^2 - 4ac.

Now, let's discuss the possible scenarios and their implications for finding the minimum value of the sum of squares of the roots:

1. If the discriminant (D) is positive, the equation has two distinct real roots. In this case, the sum of squares of the roots will be minimized when the roots are equal in magnitude but opposite in sign. For example, if the roots are x1 and x2, the sum of squares of the roots (S) will be minimized when x1 = -x2. The minimum value of S can be calculated as S = x1^2 + x2^2 = 2x1^2.

2. If the discriminant (D) is zero, the equation has two identical real roots. In this case, the sum of squares of the roots will be minimized when the roots are equal in magnitude but opposite in sign, similar to the previous scenario. The minimum value of the sum of squares of the roots can also be calculated as S = 2x1^2.

3. If the discriminant (D) is negative, the equation has two complex roots. In this case, the concept of finding the minimum value of the sum of squares of the roots may not be applicable, as the roots are not real numbers.

Please note that the above discussion is based on general principles and may not directly apply to the specific equation mentioned in the question. To provide a more accurate and specific answer, it would be helpful to have the exact equation mentioned in the question.

If you can provide the specific equation, I would be happy to assist you further in finding the minimum value of the sum of squares of the roots.