The maximum value of the expression ax² + 10x + c is 7, which means that the parabola represented by this equation opens upwards and its vertex is at the highest point.
The vertex of a parabola represented by the equation ax² + 10x + c is given by (-b/2a, f(-b/2a)), where b is the coefficient of x, a is the coefficient of x² and f(x) is the function.
Therefore the vertex of this parabola is (-10/2a, 7 - c)
Since the vertex is the highest point, the y-coordinate of the vertex is the maximum value of the expression.
7 - c = 7
therefore c = 0
We also know that the x-coordinate of the vertex is -10/2a, this means that the x-coordinate of the vertex is -5/a
There are many possible combinations of a and c that work for this question, some possible combinations are:
a=1, c=0
a=2, c=0
a=-2, c=0
Another way to find a and c is by expanding the equation:
ax² + 10x + c = a(x²+ 10/a x) + c = a(x²+ 10/a x + (100-100a)/(4a²)) + c = a(x+5/a)² + c -25/a + 7
So, when a=1, c=0 and a=2, c=0 will give you the maximum value of 7, where it's a parabola that opens upwards and it's vertex is at the highest point.