Question

graph the equation $y = x^2 + 10x + 21$ on the accompanying set of axes. you must plot 5 points including the roots and the vertex. using the graph, determine the roots of the equation $x^2 + 10x + 21 = 0$.
click to plot points. click points to delete them.

Explanation:

Step1: Find roots (y=0)

Solve $x^2 + 10x + 21 = 0$.
Factor: $(x+3)(x+7)=0$
Roots: $x=-3, x=-7$
Points: $(-7,0), (-3,0)$

Step2: Find vertex x-coordinate

Use $x=-\frac{b}{2a}$, $a=1, b=10$
$x=-\frac{10}{2(1)}=-5$

Step3: Find vertex y-coordinate

Substitute $x=-5$ into $y=x^2+10x+21$
$y=(-5)^2+10(-5)+21=25-50+21=-4$
Vertex: $(-5,-4)$

Step4: Find 2 additional points

Choose $x=-4$: $y=(-4)^2+10(-4)+21=16-40+21=-3$ → $(-4,-3)$
Choose $x=-6$: $y=(-6)^2+10(-6)+21=36-60+21=-3$ → $(-6,-3)$

Step5: Identify roots from equation

Roots are x-values where $y=0$

Answer:

The roots of $x^2 + 10x + 21 = 0$ are $\boldsymbol{x=-7}$ and $\boldsymbol{x=-3}$
(The 5 points to plot are: $(-7,0), (-6,-3), (-5,-4), (-4,-3), (-3,0)$)