Question

what are the solutions to the simple system of equations below? select all that apply.
$x^2 + 2y = -6$
$x + 8y = 0$
options: (2, 3), (-3, 2), (3, 4), no solutions

Explanation:

Step1: Isolate $x$ from linear eq.

From $x + 8y = 0$, we get $x = -8y$.

Step2: Substitute $x=-8y$ into quadratic eq.

$$(-8y)^2 + 2y = -6$$
$$64y^2 + 2y + 6 = 0$$
Simplify by dividing by 2:
$$32y^2 + y + 3 = 0$$

Step3: Check discriminant of quadratic.

Discriminant formula: $\Delta = b^2 - 4ac$
Here $a=32$, $b=1$, $c=3$:
$$\Delta = 1^2 - 4(32)(3) = 1 - 384 = -383$$

Step4: Analyze discriminant value.

Since $\Delta < 0$, no real solutions exist.

Answer:

No solutions