Question

Give all solutions if the nonlinear system of equations, including those with no real complex components.

y=x^2+6x
4x-y=-24

Answer 1

y=x^2+6x
Apply the value of y into the equation below:
4x-y=-24
4x-(x^2+6x) = -24
4x-x^2-6x = -24
x^2+2x-24=0
(x+6)(x-4)=0
Therefore x can be -6 or 4
If x = -6
Then y = (-6)^2+6(-6) = 36 -36=0
If x = 4
Then y = (4)^2 + 6(4) = 16 + 24 = 40

Answer 2

`\left\{\begin{array}{ccc}y=x^2+6x\\4x-y=-24\end{array}\right\\\\substitute:\\\\4x-(x^2+6x)=-24\\\\4x-x^2-6x+24=0\\\\-x^2-2x+24=0\\\\a=-1;\ b=-2;\ c=24\\\\\Delta=b^2-4ac`


`\Delta=(-2)^2-4\cdot(-1)\cdot24=4+96=100\\\\x_1=(-b-\sqrt\Delta)/(2a);\ x_2=(-b+\sqrt\Delta)/(2a)\\\\\sqrt\Delta=√(100)=10\\\\x_1=(2-10)/(2\cdot(-1))=(-8)/(-2)=4;\ x_2=(2+10)/(2\cdot(-1))=(12)/(-2)=-6\\\\y_1=4^2+6\cdot4=16+24=40;\ y_2=(-6)^2+6\cdot(-6)=36-36=0\\\\Answer:\\x=4\ and\ y=40\ or\ x=-6\ and\ y=0`