which represents the solution(s) of the system of equations, $y = x^{2}-4x - 21$ and $y=-5x - 22$? determine…

which represents the solution(s) of the system of equations, $y = x^{2}-4x - 21$ and $y=-5x - 22$? determine the solution set algebraically.\n(-1,-17)\n(1,-27)\n(-1,-17) and (1,-27)\nno solutions

which represents the solution(s) of the system of equations, $y = x^{2}-4x - 21$ and $y=-5x - 22$? determine the solution set algebraically.\n(-1,-17)\n(1,-27)\n(-1,-17) and (1,-27)\nno solutions

Answer

Explanation:

Step1: Set the two equations equal

Since $y = x^{2}-4x - 21$ and $y=-5x - 22$, we set $x^{2}-4x - 21=-5x - 22$.

Step2: Rearrange to standard quadratic form

Move all terms to one - side: $x^{2}-4x + 5x-21 + 22 = 0$, which simplifies to $x^{2}+x + 1 = 0$.

Step3: Use the quadratic formula

The quadratic formula for $ax^{2}+bx + c = 0$ is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Here, $a = 1$, $b = 1$, and $c = 1$. Calculate the discriminant $\Delta=b^{2}-4ac=(1)^{2}-4\times1\times1=1 - 4=-3$.

Step4: Analyze the discriminant

Since $\Delta=-3<0$, there are no real - valued solutions for $x$. And since $y$ is defined in terms of $x$ in the original equations, there are no solutions for the system of equations.

Answer:

no solutions