Question

(x^2-7)^2-4(x^2-7)-45=0

Answer 1

Ответ:

Сначала произведем замену.

Пусть у = х2 – 7.

Тогда первоначальное уравнение примет следующий вид:

У2 – 4у – 45 = 0.

Решим уравнение черед дискриминант.

D = (-4)2 – 4 * 1 * (-45) = 16 + 180 = 196 = 142.

Значит √D = 14.

Теперь вычислим корни по формуле.

у1 = (-(b) + √D) / 2a.

у2 = (-(b) - √D) / 2a.

Подставим значения и вычислим.

1) у1 = (-(-4) + 14) / 2.

у1 = (4 + 14) / 2.

у1 = 18 / 2.

у1 = 9.

2) у2 = (-(-4) - 14) / 2.

у2 = (4 - 14) / 2.

у2 = -10 / 2.

у2 = -5.

Осталось подставить полученные значения в выражение у = х2 – 7.

1) у1 = 9.

9 = х2 – 7.

х2 = 16.

Х1 = √16.

Х1 = 4.

Х2 = -√16.

Х2 = -4.

2) у2 = -5.

-5 = х2 – 7.

х2 = 2.

Х3 = √2.

Х4 = -√2.

Ответ: Х1 = 4, Х2 = -4, Х3 = √2, Х4 = -√2.

Объяснение:

Answer 2

To solve the quadratic equation $(x^2 - 7)^2 - 4(x^2 - 7) - 45 = 0$, you can use a substitution to simplify the equation. Let $y = x^2 - 7$, so the equation becomes:

$y^2 - 4y - 45 = 0$

Now, you have a quadratic equation in terms of $y$, which is easier to solve. To solve it, you can factor it or use the quadratic formula.

Factor the equation:

$(y - 9)(y + 5) = 0$

Now, set each factor equal to zero:

1. $y - 9 = 0$ $y = 9$

2. $y + 5 = 0$ $y = -5$

Now that you have the values of $y$, you can substitute back the expression for $y$ in terms of $x$:

For $y = 9$: $x^2 - 7 = 9$

Add 7 to both sides: $x^2 = 16$

Take the square root of both sides: $x = \pm 4$

For $y = -5$: $x^2 - 7 = -5$

Add 7 to both sides: $x^2 = 2$

Take the square root of both sides: $x = \pm \sqrt{2}$

So, the solutions to the original equation $(x^2 - 7)^2 - 4(x^2 - 7) - 45 = 0$ are $x = 4$, $x = -4$, $x = \sqrt{2}$, and $x = -\sqrt{2}$.