First, we can write the quadratic function in the form:
ax^2 + bx + c = a(x^2 + (b/a)x) + c
ax^2 + bx + c = a(x^2 + (b/a)x + (b/2a)^2 - (b/2a)^2) + c
ax^2 + bx + c = a[(x + (b/2a))^2 - (b/2a)^2] + c
minimum value of this expression occurs when (x + (b/2a))^2 = 0, which is only possible when x = -(b/2a)
ax^2 + bx + c = a(0 - (b/2a)^2) + c = -b^2/4a + c
the minimum value of the quadratic function is -(Δ/4a), which is equivalent to -b^2/4a when a > 0
the function is zero when x = 1, so we can write:
a(1)^2 + b(1) + c = 0
a + b + c = 0
ax^2 + bx + c = a(x - h)^2 + k, where h = -b/2a and k = -b^2/4a + c
the value of the function at x = 0 is 5, so we have:
a(0)^2 + b(0) + c = 5
c = 5
k = -b^2/4a + c
k = -(-a-5)^2/4a + 5
Simplifying this expression, we get:
k = (-a^2 - 10a - 25)/4a + 5
k = (-a^2 - 10a + 15)/4a
Since we know that k = -4, we can write:
-4 = (-a^2 - 10a + 15)/4a
Multiplying both sides by 4a, we get:
-16a = -a^2 - 10a + 15
Simplifying this equation, we get:
a^2 - 6a - 15 = 0
Factoring this quadratic equation, we get:
(a - 5)(a + 3) = 0
So, either a = 5 or a = -3. If a = 5, we can solve for b using the equation a + b
*IG:whis.sama_ent