Question

find all solutions of the system of equations algebraically. write your solutions as coordinate points.

$y = -2x^2 - 5x - 3$
$3 = x - y$

Explanation:

Step1: Isolate y from linear equation

From $3 = x - y$, rearrange to solve for $y$:
$y = x - 3$

Step2: Substitute y into quadratic equation

Replace $y$ in $y = -2x^2 -5x -3$ with $x-3$:
$x - 3 = -2x^2 -5x -3$

Step3: Simplify to standard quadratic form

Bring all terms to one side:
$2x^2 +6x = 0$
Factor out $2x$:
$2x(x + 3) = 0$

Step4: Solve for x values

Set each factor equal to 0:
$2x = 0 \implies x=0$
$x+3=0 \implies x=-3$

Step5: Find corresponding y values

For $x=0$: $y = 0 - 3 = -3$
For $x=-3$: $y = -3 - 3 = -6$

Answer:

$(0, -3)$ and $(-3, -6)$