Question

Решите уравнение (х^2+8x)^2 - 4(x+4)^2=256,используя метод введения новой переменной

Answer 1

(х^2+8х)^2-4(х+4)^2=256 {-10,-4,-4.2}

Answer 2

X^4+16x^3+64x^2 - 4x^2-32x-64=256  => x1= -10,x2= 2,x3= -4

Answer 3

Solving the Equation using the Method of Introducing a New Variable

To solve the equation *(x^2 + 8x)^2 - 4(x+4)^2 = 256* using the method of introducing a new variable, we can follow these steps:

1. Let's introduce a new variable *y* such that *y = (x^2 + 8x)*. 2. Then the given equation becomes *y^2 - 4(x+4)^2 = 256*.

Now, we can solve for *y* and then substitute back to find the values of *x*.

Substituting the New Variable

Let's substitute *y = (x^2 + 8x)* into the equation:

*(x^2 + 8x)^2 - 4(x+4)^2 = 256* becomes *y^2 - 4(x+4)^2 = 256*.

Solving for y

Now, we can solve for *y* using the equation *y^2 - 4(x+4)^2 = 256*.

*y^2 - 4(x+4)^2 = 256* can be rewritten as *y^2 = 4(x+4)^2 + 256*.

Simplifying the Equation

Simplify the equation by expanding the terms and solving for *y*.

*y^2 = 4(x+4)^2 + 256* becomes *y^2 = 4(x^2 + 8x + 16) + 256*, *y^2 = 4x^2 + 32x + 64 + 256*, *y^2 = 4x^2 + 32x + 320*.

Solving for x

Now that we have the equation *y^2 = 4x^2 + 32x + 320*, we can solve for *x*.

This is a quadratic equation in terms of *x*, and we can solve for *x* using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a), where the coefficients are: - a = 4 - b = 32 - c = 320

Conclusion

By solving the quadratic equation for *x*, we can find the values of *x* that satisfy the original equation *(x^2 + 8x)^2 - 4(x+4)^2 = 256* using the method of introducing a new variable.