Question

X4 + 7x2 -18 = 0 Через биквадратное решение

Answer 1

Пусть x²=a
а²+7а-18=0
Д =(7)² -4*1*(-18)= 49+72=121
√Д=√121=11
а1=(-7+11)/2=4/2=2
а2=(-7-11)/2=-18/2=-9
Подставляем:
1) x²=2
    x=-√2
    x=√2
2)x²=-9 - не может быть

Ответ: x=-√2
            x=√2

Answer 2

Solving the Quadratic Equation: x^4 + 7x^2 - 18 = 0

To solve the equation x^4 + 7x^2 - 18 = 0 using the biquadratic method, we can substitute a new variable to simplify the equation. Let's use y = x^2.

Substituting y into the equation, we get y^2 + 7y - 18 = 0. Now, we can solve this quadratic equation for y using factoring, completing the square, or the quadratic formula.

Let's use the quadratic formula to find the values of y:

The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

In our case, a = 1, b = 7, and c = -18. Plugging these values into the quadratic formula, we get:

y = (-7 ± √(7^2 - 4(1)(-18))) / (2(1))

Simplifying further:

y = (-7 ± √(49 + 72)) / 2

y = (-7 ± √121) / 2

y = (-7 ± 11) / 2

This gives us two possible values for y:

1. When y = (-7 + 11) / 2 = 2 2. When y = (-7 - 11) / 2 = -9

Now that we have the values of y, we can substitute them back into the equation y = x^2 to find the corresponding values of x.

1. When y = 2: x^2 = 2 Taking the square root of both sides, we get: x = ±√2

2. When y = -9: x^2 = -9 Since the square root of a negative number is not a real number, there are no real solutions for x in this case.

Therefore, the solutions to the equation x^4 + 7x^2 - 18 = 0 using the biquadratic method are: x = ±√2

Please note that the biquadratic method is one of the approaches to solve this equation, and there may be alternative methods as well.