Question

X^4+2006x^2-2007=0....................................

Answer 1

X^2 = A
A^2 + 2006A - 2007 = 0
D = 4024036 - 4*(-2007) = 4024036 + 8028 = 4032064   ( V D = 2008)
A1 = ( - 2006 + 2008) \ 2 = 1
A2 = ( - 4014) \ 2 = ( - 2007)
т.к. X^2 = A ( A > 0)
X^2 = 1 ---> A = 1
ОТВЕТ: А = 1

Answer 2

To solve the quadratic equation $x^4 + 2006x^2 - 2007 = 0$, we can substitute $y = x^2$ to convert it into a quadratic equation.

Let's rewrite the equation with this substitution:

$y^2 + 2006y - 2007 = 0$

Now we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula in this case:

The quadratic formula states that for an equation in the form $ay^2 + by + c = 0$, the solutions for $y$ can be found using the formula:

$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For our equation $y^2 + 2006y - 2007 = 0$, we have $a = 1$, $b = 2006$, and $c = -2007$. Let's substitute these values into the quadratic formula to find the solutions for $y$:

$y = \frac{-2006 \pm \sqrt{2006^2 - 4 \cdot 1 \cdot (-2007)}}{2 \cdot 1}$

Simplifying this expression gives us:

$y = \frac{-2006 \pm \sqrt{4024036 + 8028}}{2}$

$y = \frac{-2006 \pm \sqrt{4032064}}{2}$

$y = \frac{-2006 \pm 2016}{2}$

Now we have two possibilities for $y$:

1. $y = \frac{-2006 + 2016}{2} = \frac{10}{2} = 5$

2. $y = \frac{-2006 - 2016}{2} = \frac{-4022}{2} = -2011$

Now we substitute the values of $y$ back into the equation $y = x^2$:

1. $x^2 = 5$

Taking the square root of both sides gives us:

$x = \pm \sqrt{5}$

2. $x^2 = -2011$

This equation has no real solutions because the square of a real number cannot be negative. Therefore, the equation $x^4 + 2006x^2 - 2007 = 0$ has two real solutions: $x = \sqrt{5}$ and $x = -\sqrt{5}$.