Question 515337
-(x+5)^2 -6 = -(x+5)(x+5) -6 = -(x^2 +10x + 25) -6
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-(x^2 +10x + 25) -6 = -x^2 -10x -25 -6 = -x^2 -10x -31
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The graph will help orient us to the solution
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{{{ graph(500,500,-12,5,-12,5,-x^2 -10*x -31) }}}
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The vertex has a x value of -b/2a = -(-10)/(-2*1) = -5
When x=-5
y = -x^2 -10x -31
y = -25 +50 -31
y = -6
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The vertex = (-5,-6).
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The line of symmetry is x=-5, a vertical line.
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Since the parabola opens 'down' we know the vertex is the maximum.
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We also know this from the graph.
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And we can apply calculus to set the first derivative = 0.
dy/dx = -2x -10 = 0
2x = -10
x = -5
Correct.
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Done.