This is not an equation to solve but rather represents the graph of a parabola.
What we can do is to work out the shape so the graph can be drawn.
y=-5x2+114x+23=-(5x2-114x-23)=-(x-23)(5x+1), so the x-intercepts are at 23 and -⅕ or -0.2.
If we can get this into vertex form we can find where the vertex is. At this stage we can see that the y-intercept is 23 because when x=0 (the y-axis) y=23, so the parabola crosses the y-axis at y=23. Also we know the parabola has to pass through the x-intercepts at (23,0) and (-0.2,0).
We can also see that the sign in front of x2 is negative which means that the parabola is shaped like an inverted U and one of its arms passes through (0,23).
y=-5(x2-114x/5-23/5) is another way of writing the equation.
Now we need to complete the square x2-114x/5, so we divide the x coefficient by 2 and square the result:
(-57/5)2=3249/25. Add this number: x2-114x/5+3249/25=(x-57/5)2.
But we have to balance the equation by also subtracting 3249/25:
y=-5(x2-114x/5+3249/25-3249/25-23/5)=-5((x-57/5)2+(-23/5-3249/25)); -23/5-3249/25=-(115+3249)/25=-3364/25.
y=-5(x-57/5)2-5(-3364/25),
y=-5(x-57/5)2+3364/5.
We can see from this that, when x=57/5, y=3364/5 which is the maximum value for y because all other values of x will subtract from this. So (57/5,3364/5) is the vertex.
57/5=11.4 and 3364/5=672.8. The vertex is at (11.4,672.8). This shows that it's the parabola's left arm which passes through the y-axis at y=23.
From all the information above we can see that the parabola is quite narrow because the vertex is 672.8 above the x-axis, but the distance between the two x-intercepts is only 23.2.
This provides enough information to sketch the parabola. Below is the graph. Compare the scales of the two axes to accommodate the parabola.
