Question

При каких значениях параметра a сумма квадратов корней уравнения x^2+ax=2x+a+3 равна 18?

Answer 1

Х^2+(а-2)*х-(а+3)=0; х1*х2=-(а+3) х1+х2=-(а-2)=(2-а); (х1+х2)^2=(х1)^2+(х2)^2+2*х1*х2; (х1)^2+(х2)^2=(х1+х2)^2-2*х1*х2; 18=(2-а)^2+2*(а+3); 4-4а+а^2+2а+6-18=0; а^2-2а-8=0; D=36; a=(2+6):2=4 и а=(2-6):2=-2; ответ: -2 и 4

Answer 2

Solution for Finding the Values of Parameter "a"

To find the values of the parameter "a" for which the sum of the squares of the roots of the equation \(x^2 + ax = 2x + a + 3\) is equal to 18, we can use the following approach:

1. Identify the Equation: - The given equation is \(x^2 + ax = 2x + a + 3\).

2. Determine the Sum of Squares of Roots: - The sum of the squares of the roots of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(\alpha^2 + \beta^2 = (a^2 - 2ac)\), where \(\alpha\) and \(\beta\) are the roots of the equation.

3. Apply the Formula: - We can apply the formula to the given equation to find the values of "a" for which the sum of the squares of the roots is equal to 18.

4. Solve for "a": - By comparing the coefficients of the given equation with the standard form of a quadratic equation, we can solve for "a" to satisfy the condition.

Application of the Formula

Using the formula for the sum of the squares of the roots, we can apply it to the given equation \(x^2 + ax = 2x + a + 3\) to find the values of "a" for which the sum of the squares of the roots is equal to 18.

The formula for the sum of the squares of the roots is \(\alpha^2 + \beta^2 = (a^2 - 2ac)\).

Solving for "a"

By comparing the coefficients of the given equation with the standard form of a quadratic equation, we can solve for "a" to satisfy the condition that the sum of the squares of the roots is equal to 18.

Upon solving, the values of "a" for which the sum of the squares of the roots of the equation \(x^2 + ax = 2x + a + 3\) is equal to 18 are within the range of \([4, +\infty)\).

Therefore, the values of the parameter "a" for which the sum of the squares of the roots of the equation \(x^2 + ax = 2x + a + 3\) is equal to 18 are within the range of [4, +∞).