Think of two numbers that
Trial and error will get us to the values 2 and 5
This would mean x^2+7x+10 = (x+2)(x+5)
Set each factor equal to zero and solve for x.
- x+2 = 0 solves to x = -2
- x+5 = 0 solves to x = -5
The roots or x intercepts are -2 and -5, which is where the parabola crosses the x axis. This represents the locations (-2,0) and (-5,0) respectively.
Side note: The quadratic formula can be used to solve x^2+7x+10 = 0 as an alternative route.
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The roots found were -2 and -5.
The x coordinate of the vertex is found at the midpoint of these roots.
Add them up and divide in half
(-2 + -5)/2 = -7/2 = -3.5
Plug this value into the function to find the y coordinate of the vertex.
f(x) = x^2+7x+10
f(-3.5) = (-3.5)^2+7(-3.5)+10
f(-3.5) = -2.25
The vertex is located at (-3.5, -2.25)
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In conclusion we have these three points on the parabola
- (-2,0) = x intercept
- (-5,0) = x intercept
- (-3.5, -2.25) = vertex
Check out the graph below. I used GeoGebra to make the graph, but Desmos is another good option.