Question

(x^2+5x+1)×(x^2+5x+3)+1=0

Answer 1

Ответ:

$\dfrac{-5 \pm \sqrt{17}}{2}$

Пошаговое объяснение:

$(x^{2}+5x+1)(x^{2}+5x+3)+1=0;$

$(x^{2}+5x+2-1)(x^{2}+5x+2+1)+1=0;$

$(x^{2}+5x+2)^{2}-1^{2}+1=0;$

$(x^{2}+5x+2)^{2}-1+1=0;$

$(x^{2}+5x+2)^{2}=0;$

$x^{2}+5x+2=0;$

$D=b^{2}-4ac \Rightarrow D=5^{2}-4 \cdot 1 \cdot 2=25-8=17;$

$x_{1,2}=\dfrac{-b \pm \sqrt{D}}{2a} \Rightarrow x_{1,2}=\dfrac{-5 \pm \sqrt{17}}{2 \cdot 1}=\dfrac{-5 \pm \sqrt{17}}{2};$

Answer 2

To solve the equation $(x^2+5x+1) \cdot (x^2+5x+3) + 1 = 0$, you can follow these steps:

1. Expand the product: $(x^2+5x+1) \cdot (x^2+5x+3) = x^4 + 5x^3 + 3x^2 + 5x^3 + 25x^2 + 15x + x^2 + 5x + 3$

2. Combine like terms: $x^4 + 10x^3 + 29x^2 + 20x + 3$

3. Now, the equation becomes: $x^4 + 10x^3 + 29x^2 + 20x + 3 + 1 = 0$

4. Simplify: $x^4 + 10x^3 + 29x^2 + 20x + 4 = 0$

Now, you have a quartic equation. Solving quartic equations can be complex and often involve numerical methods or specialized software. If you want to find the exact roots, you may use a computer algebra system or numerical approximation methods like the Newton-Raphson method.