Question

Дана функция f(x)=(a-2)x²-a²x+6 Решите уравнение f(x)=f(1), если известно, что f(-2)=-4

Answer 1

F(x) = (a-2)*x^2 - a^2*x + 6
f(-2) = (a-2)*4 - a^2*(-2) + 6 = 4a - 8 + 2a^2 + 6 = 2a^2 + 4a - 2 = -4
2a^2 + 4a + 2 = 0
2(a^2 + 2a + 1) = 2(a + 1)^2 = 0
a = -1
Теперь подставляем в функцию.
f(x) = (-1-2)*x^2 - (-1)^2*x + 6 = -3x^2 - x + 6
f(1) = -3*1^2 - 1 + 6 = -3 - 1 + 6 = 2
Теперь решаем уравнение
f(x) = f(1)
-3x^2 - x + 6 = 2
3x^2 + x - 4 = 0
(x - 1)(3x + 4) = 0
x1 = 1; x2 = -4/3

Answer 2

Given Information:

We are given the function f(x) = (a-2)x² - a²x + 6 and we need to solve the equation f(x) = f(1), given that f(-2) = -4.

Step 1: Calculate f(x)

To solve the equation f(x) = f(1), we first need to find the value of f(x). Substituting x = -2 in the given function, we get: f(-2) = (a-2)(-2)² - a²(-2) + 6 = -4

Simplifying the equation, we have: 4(a-2) + 4a² + 6 = -4

Step 2: Solve for a

Now, we can solve the equation obtained in Step 1 to find the value of a.

Expanding the equation, we have: 4a - 8 + 4a² + 6 = -4

Combining like terms, we get: 4a² + 4a - 6 = -4 + 8

Simplifying further, we have: 4a² + 4a - 6 = 4

Moving all terms to one side, we get: 4a² + 4a - 10 = 0

Step 3: Solve the quadratic equation

To solve the quadratic equation 4a² + 4a - 10 = 0, we can use the quadratic formula: a = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 4, b = 4, and c = -10. Substituting these values into the quadratic formula, we have: a = (-4 ± √(4² - 4 * 4 * -10)) / (2 * 4)

Simplifying further, we get: a = (-4 ± √(16 + 160)) / 8 a = (-4 ± √(176)) / 8 a = (-4 ± 4√(11)) / 8 a = (-1 ± √(11)) / 2

Therefore, the solutions for a are: a₁ = (-1 + √(11)) / 2 a₂ = (-1 - √(11)) / 2

Step 4: Substitute a values into the equation f(x) = f(1)

Finally, substitute the values of a obtained in Step 3 back into the equation f(x) = f(1) to find the values of x.

For a = (-1 + √(11)) / 2: f(x) = ((-1 + √(11)) / 2 - 2)x² - ((-1 + √(11)) / 2)²x + 6

For a = (-1 - √(11)) / 2: f(x) = ((-1 - √(11)) / 2 - 2)x² - ((-1 - √(11)) / 2)²x + 6

Solving these equations will give us the values of x that satisfy f(x) = f(1).