Question

Когда свободный член квадратного трехчлена умножили на 16, его дискриминант также умножился на 16. Какой наименьший корень может иметь получившийся трехчлен, если один из корней исходного трехчлена равен 3?

Answer 1

Ответ:

-12

Пошаговое объяснение:
Рассмотрим квадратное уравнение вида:
ax² + bx + c = 0

где a,b,c - коэффициенты.

Определение дискриминанта:
D = b² - 4ac

Если мы умножим свободный член c на 16, должно получиться 16с ⇒ ax² + bx + 16c = 0 ⇒ D = b² - 4a(16c) = b² - 64ac

Согласно условию задачи, дискриминант нового уравнения в 16 раз больше дискриминанта исходного уравнения ⇒ D=16D

b² - 64ac = 16(b² - 4ac).

b² - 64ac = 16b² - 64ac. ⇒ b² = 0 ⇒ ax² + 16c = 0

$3= \frac{-b+\sqrt{D} }{2a} = \frac{\sqrt{D} }{2a} \\\\\sqrt{D} =6a\\\\x_{1,2} =+-\frac{2*6a}{a} = +-12$

Ответ: -12

Answer 2

Understanding the Problem

The problem states that when the constant term of a quadratic trinomial is multiplied by 16, the discriminant of the trinomial is also multiplied by 16. We are asked to find the smallest possible root of the resulting trinomial if one of the roots of the original trinomial is 3.

Solving the Problem

To solve this problem, let's start by understanding the relationship between the constant term, the discriminant, and the roots of a quadratic trinomial.

A quadratic trinomial can be written in the form: ax^2 + bx + c = 0, where a, b, and c are constants.

The discriminant (D) of a quadratic trinomial is given by the formula: D = b^2 - 4ac. The discriminant determines the nature of the roots of the trinomial.

If the discriminant is positive (D > 0), the trinomial has two distinct real roots. If the discriminant is zero (D = 0), the trinomial has one real root (a repeated root). If the discriminant is negative (D < 0), the trinomial has no real roots (two complex roots).

In this problem, we are given that one of the roots of the original trinomial is 3. Let's denote the other root as x.

Since the trinomial has two roots, we can write it in factored form as: (x - 3)(x - r) = 0, where r is the other root.

Expanding this equation, we get: x^2 - (3 + r)x + 3r = 0.

According to the problem, when the constant term (3r) is multiplied by 16, the discriminant is also multiplied by 16.

Let's denote the constant term of the resulting trinomial as k. Therefore, we have: 16k = (3 + r)^2 - 4(3r).

Simplifying this equation, we get: 16k = 9 + 6r + r^2 - 12r.

Combining like terms, we have: 16k = r^2 - 6r + 9.

Now, we need to find the smallest possible value of r that satisfies this equation.

To find the smallest value of r, we can use the fact that the discriminant is zero when the trinomial has one repeated root.

Setting the discriminant equal to zero, we have: 0 = (-6)^2 - 4(1)(9).

Simplifying this equation, we get: 0 = 36 - 36.

Therefore, the discriminant is zero when r = 3.

Substituting r = 3 into the equation 16k = r^2 - 6r + 9, we get: 16k = 3^2 - 6(3) + 9.

Simplifying this equation, we have: 16k = 9 - 18 + 9.

Combining like terms, we get: 16k = 0.

Therefore, the constant term of the resulting trinomial is 0.

Now, let's find the roots of the resulting trinomial.

Substituting r = 3 and k = 0 into the equation x^2 - (3 + r)x + 3r = 0, we get: x^2 - (3 + 3)x + 3(3) = 0.

Simplifying this equation, we have: x^2 - 6x + 9 = 0.

Factoring this equation, we get: (x - 3)(x - 3) = 0.

Therefore, the resulting trinomial has one repeated root, which is x = 3.

Answer

The smallest possible root of the resulting trinomial, when one of the roots of the original trinomial is 3, is x = 3.