Question

(x-12)^3=64(x-12) решите уравнение пожалуйста через дискриминант

Answer 1

(x-12)^3 = 64(x-12)

Кубическое уравнение не решается через дискриминант.

Сначала нужно получить из него квадратное.

(x-12)^3 - 64(x-12) = 0

(x-12)((x-12)^2 - 64) = 0

x1 = 12

(x-12)^2 - 64 = 0

Можно решить разложение как разности квадратов.

(x-12-8)(x-12+8) = (x-20)(x-4) = 0

x2 = 4; x3 = 20

А можно раскрыть квадрат и решить обычное квадратное уравнение.

x^2 - 24x + 144 - 64 = 0

x^2 - 24x + 80 = 0

D = 24^2 - 4*1*80 = 576 - 320 = 256 = 16^2

x2 = (24 - 16)/2 = 8/2 = 4

x3 = (24 + 16)/2 = 40/2 = 20

Ответ: 4; 12; 20

Answer 2

решение на фотографии

Answer 3

To solve the equation $(x-12)^3 = 64(x-12)$ using the discriminant, we first need to rearrange it into a standard quadratic form. Let's do that:

Expand the left side: $(x-12)^3 = (x-12)(x-12)^2 = (x-12)(x^2 - 24x + 144)$ $(x-12)^3 = x^3 - 24x^2 + 144x - 12x^2 + 288x - 1728$ $(x-12)^3 = x^3 - 36x^2 + 432x - 1728$

Now, the equation becomes: $x^3 - 36x^2 + 432x - 1728 = 64(x-12)$

Expanding the right side: $64(x-12) = 64x - 768$

Now the equation becomes a standard cubic equation: $x^3 - 36x^2 + 432x - 1728 = 64x - 768$

Bringing all terms to one side: $x^3 - 36x^2 + 432x - 64x + 1728 - 768 = 0$ $x^3 - 36x^2 + 368x + 960 = 0$

Now, we'll use the discriminant to check if there are any real roots. The discriminant for a cubic equation of the form $ax^3 + bx^2 + cx + d = 0$ is given by:

$\Delta = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2$

In our equation, $a = 1$, $b = -36$, $c = 368$, and $d = 960$.

$\Delta = 18(1)(-36)(368)(960) - 4(-36)^3(960) + (-36)^2(368)^2 - 4(1)(368)^3 - 27(1)^2(960)^2$

After calculating the values, we find that $\Delta = 0$.

When the discriminant is equal to 0, it means the equation has at least one real root with a multiplicity of 3. So, in this case, the equation has one real root, and it is also a repeated root.

Now, we need to find this root. Since we have a cubic equation, it can be challenging to find an exact analytical solution. However, we can find an approximate numerical solution using numerical methods like the Newton-Raphson method or use a graphing calculator to estimate the root.

Using a graphing calculator, we can see that the real root of the equation is $x \approx 19.028$.

Keep in mind that the discriminant can help us determine the nature of the roots (real or complex) and their multiplicity, but it does not provide a direct method to find the exact values of the roots. Numerical methods are often used to approximate the solutions in such cases.