Question

use a graphing calculator to solve the system of equations.
$y = -2x^2 - 12x - 16$
$y = x + 3$
round to the nearest hundredth.
if there is more than one solution, use the \or\ button.
if there is no real solution, use the
o solution\ button.
$(x, y) = (\square, \square)$

Explanation:

Step1: Set equations equal

$x + 3 = -2x^2 - 12x - 16$

Step2: Rearrange to standard quadratic

$2x^2 + 13x + 19 = 0$

Step3: Apply quadratic formula

For $ax^2+bx+c=0$, $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Here $a=2$, $b=13$, $c=19$:
$$x=\frac{-13\pm\sqrt{13^2-4(2)(19)}}{2(2)}=\frac{-13\pm\sqrt{169-152}}{4}=\frac{-13\pm\sqrt{17}}{4}$$

Step4: Calculate x-values

$\sqrt{17}\approx4.123$, so:
$x_1=\frac{-13+4.123}{4}\approx\frac{-8.877}{4}\approx-2.22$
$x_2=\frac{-13-4.123}{4}\approx\frac{-17.123}{4}\approx-4.28$

Step5: Find corresponding y-values

For $x\approx-2.22$, $y=-2.22+3=0.78$
For $x\approx-4.28$, $y=-4.28+3=-1.28$

Answer:

$(x,y)=(-2.22, 0.78)$ or $(x,y)=(-4.28, -1.28)$