Question

При каких значениях параметра р уравнение х^2 - 2(р-1)х +4р^2=0 имеет не более одного корня?? Помогите пожалуйста!!!

Answer 1

Если дискриминант D≤0 , то уравнение имеет не более одного решения.

D = 4(p-1)² - 16р² = 4p² - 8p + 4 - 16р² = -12р² - 8p + 4

-12р² - 8p + 4 ≤ 0

или

-3р² - 2p + 1 ≤ 0 (А)

Найдём корни уравнения

-3р² - 2p + 1 = 0

D = 4+12 = 16

p₁ = (2 + 4):(-6) = -1

p₂ = (2 - 4):(-6) = 1/3

Решение неравенства (А) таково: х∈(-∞, -1] и [1/3, +∞)

Это и будет ответом.

Answer 2

Finding the Values of Parameter p for the Equation to Have at Most One Root

To determine the values of the parameter p for which the equation x^2 - 2(p-1)x + 4p^2 = 0 has at most one root, we can use the discriminant of the quadratic equation.

The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac. The number of roots of the equation depends on the value of the discriminant:

1. If D > 0, the equation has two distinct real roots. 2. If D = 0, the equation has one real root (a double root). 3. If D < 0, the equation has no real roots.

In our case, the equation is x^2 - 2(p-1)x + 4p^2 = 0. Comparing this with the general form ax^2 + bx + c = 0, we have a = 1, b = -2(p-1), and c = 4p^2.

To find the values of p for which the equation has at most one root, we need to find the values of p for which the discriminant D is either equal to zero or less than zero.

Let's calculate the discriminant and solve for p:

D = (-2(p-1))^2 - 4(1)(4p^2) D = 4(p^2 - 2p + 1) - 16p^2 D = 4p^2 - 8p + 4 - 16p^2 D = -12p^2 - 8p + 4

To find the values of p for which the equation has at most one root, we need to solve the inequality D ≤ 0:

-12p^2 - 8p + 4 ≤ 0

Now, let's solve this inequality to find the values of p:

-12p^2 - 8p + 4 ≤ 0 3p^2 + 2p - 1 ≥ 0

To solve this quadratic inequality, we can factor it or use the quadratic formula. However, since the question asks for a detailed response, we will use the quadratic formula to find the roots of the equation 3p^2 + 2p - 1 = 0:

p = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 3, b = 2, and c = -1. Substituting these values into the quadratic formula, we get:

p = (-2 ± √(2^2 - 4(3)(-1))) / (2(3)) p = (-2 ± √(4 + 12)) / 6 p = (-2 ± √16) / 6 p = (-2 ± 4) / 6

Simplifying further, we have:

p = (2 - 4) / 6 or p = (2 + 4) / 6 p = -2/6 or p = 6/6 p = -1/3 or p = 1

Therefore, the values of p for which the equation x^2 - 2(p-1)x + 4p^2 = 0 has at most one root are p = -1/3 and p = 1.

Note: Please double-check the calculations as they were done manually.