Remember that in the Quadratic Formula the discriminant (the part under the square root ), b^2 - 4ac, gives us some info about the solutions we can expect
If the discriminant evaluates to 0, it means that we have a "double root", i.e., only one solution
And since we only want one solution point in this problem - the tangent point of the line to the parabola - we can see what value of "b" gives us a discriminant of 0
So... using x^2 - 4x + 7 - b = 0 let's change "b" to "m" so that we don't get it confused wih the "b" in the quadratic formula
So we have x^2 - 4x + 7 - m = 0
And in the quadratic formula, let a = 1, b = -4 and c = 7 - m
So.... the discriminant becomes b^2 - 4ac → (-4)^2 - 4(1)(7 - m)
Now....set this to 0 and solve for "m"
(-4)^2 - 4(1)(7 - m) = 0 simplify
16- 28 + 4m = 0
-12 + 4m = 0
4m = 12
m = 3
So.....the "m" - or in this case, the "b" - that gives us a single solution is 3