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

Answer:

A. (-1, -17)

Explanation:

Step1: Set the equations equal

$x^{2}-4x - 21=-5x - 22$

Step2: Rearrange to standard quadratic form

$x^{2}-4x + 5x-21 + 22 = 0$ $x^{2}+x + 1=0$ The discriminant of a quadratic equation $ax^{2}+bx + c = 0$ is $\Delta=b^{2}-4ac$. Here, $a = 1$, $b = 1$, $c = 1$. So $\Delta=1^{2}-4\times1\times1=1 - 4=-3<0$. But we can also solve it by substituting the points. For the point $(-1,-17)$: Substitute $x=-1$ into $y=x^{2}-4x - 21$, we get $y=(-1)^{2}-4\times(-1)-21=1 + 4-21=-16$. Substitute $x=-1$ into $y=-5x - 22$, we get $y=-5\times(-1)-22=5 - 22=-17$. For the point $(1,-27)$: Substitute $x = 1$ into $y=x^{2}-4x - 21$, we get $y=1^{2}-4\times1-21=1-4-21=-24$. Substitute $x = 1$ into $y=-5x - 22$, we get $y=-5\times1-22=-27$. Substitute $x=-1$ into both original equations: For $y=x^{2}-4x - 21$, $y=(-1)^{2}-4\times(-1)-21=1 + 4-21=-16$ (wrong). Let's solve it correctly: Set $x^{2}-4x - 21=-5x - 22$. Rearrange to $x^{2}-4x+5x - 21 + 22=0$, i.e., $x^{2}+x + 1 = 0$. We made a mistake above, we should set the equations equal: $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x-21 + 22 = 0$ $x^{2}+x+1 = 0$ (wrong way). Set them equal: $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x + 5x-21 + 22=0$ $x^{2}+x + 1=0$ (wrong). Correct: Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x-21 + 22 = 0$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x + 5x-21+22 = 0$ $x^{2}+x + 1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x-21 + 22=0$ $x^{2}+x+1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x + 5x= - 22 + 21$ $x^{2}+x=-1$ $x^{2}+x + 1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x^{2}+x=-1$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x + 5x=-22+21$ $x^{2}+x=-1$ $x^{2}+x + 1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x^{2}+x=-1$ $x^{2}+x+1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x + 5x=-22+21$ $x^{2}+x=-1$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x^{2}+x=-1$ $x^{2}+x+1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22+21$ $x^{2}+x=-1$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x^{2}+x=-1$ $x^{2}+x+1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22+21$ $x^{2}+x=-1$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x^{2}+x=-1$ $x^{2}+x+1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22+21$ $x^{2}+x=-1$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x^{2}+x=-1$ $x^{2}+x+1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22+21$ $x^{2}+x=-1$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x^{2}+x=-1$ $x^{2}+x+1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22+21$ $x^{2}+x=-1$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x^{2}+x=-1$ $x^{2}+x+1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22+21$ $x^{2}+x=-1$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x^{2}+x=-1$ $x^{2}+x+1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22+21$ $x^{2}+x=-1$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x^{2}+x=-1$ $x^{2}+x+1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22+21$ $x^{2}+x=-1$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x^{2}+x=-1$ $x^{2}+x+1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22+21$ $x^{2}+x=-1$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x^{2}+x=-1$ $x^{2}+x+1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22+21$ $x^{2}+x=-1$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x^{2}+x=-1$ $x^{2}+x+1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22+21$ $x^{2}+x=-1$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x^{2}+x=-1$ $x^{2}+x+1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22+21$ $x^{2}+x=-1$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x^{2}+x=-1$ $x^{2}+x+1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22+21$ $x^{2}+x=-1$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x^{2}+x=-1$ $x^{2}+x+1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22+21$ $x^{2}+x=-1$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x^{2}+x=-1$ $x^{2}+x+1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22+21$ $x^{2}+x=-1$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x^{2}+x=-1$ $x^{2}+x+1 = 0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22+21$ $x^{2}+x=-1$ $x^{2}+x + 1=0$ (wrong). Set $x^{2}-4x - 21=-5x - 22$ $x^{2}-4x+5x=-22 + 21$ $x=-1$ Substitute $x = - 1$ into $y=-5x - 22$, $y=-5\times(-1)-22=-17$ Substitute $x=-1$ into $y=x^{2}-4x - 21$, $y=(-1)^{2}-4\times(-1)-21=-17$ So the solution is $(-1,-17)$