Answer:
42.875
Step-by-step explanation:
First you find the x-intercepts and the vertex of the equation.
(-5, 0), (-2, 0), (-3/2, 49/4)
You use those coordinates to find the sides of the triangle and then you use the lengths to find the semiperimeter.
Lastly, you use Heron's formula to compute the area.
The area of the triangle is required.
The area of the triangle is [tex]42.875\ \text{units}^2[/tex]
[tex]y=-x^2-3x+10[/tex]
The x intercepts are
[tex]-x^2-3x+10=0\\\Rightarrow x=\dfrac{-\left(-3\right)\pm \sqrt{\left(-3\right)^2-4\left(-1\right)\times 10}}{2\left(-1\right)}\\\Rightarrow x=-5,2[/tex]
The x intercepts are
[tex](-5,0),(2,0)[/tex]
Distance between them is
[tex]2-(-5)=7\ \text{units}[/tex]
Vertex of a parabola is given by
[tex]x=-\dfrac{b}{2a}\\ =-\dfrac{-3}{2\times -1}\\ =-1.5[/tex]
[tex]y=-(-1.5)^2-3(-1.5)+10=12.25[/tex]
So,
Base = b = 7 units
Height = h = 12.25 units
Area is
[tex]A=\dfrac{1}{2}bh\\\Rightarrow A=\dfrac{1}{2}\times 7\times 12.25\\\Rightarrow A=42.875\ \text{units}^2[/tex]
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