A quadratic equation has x = -2 + 3i as a solution and passes through the point (-1, -15). Write the quadratic equation in standard form.

The standard form of the quadratic equation is f(x) = 2/3·(x + 2)² + 6

Step-by-step explanation:

The given solution of the quadratic equation is x = -2 + 3·i

A point on the path of the quadratic equation = (-1, -15)

The general form of the quadratic equation is f(x) = a·x² + b·x + c

When f(x) = 0, (At which the solution is found), we have x = -2 + 3·i

Substituting gives;

0 = a·(-2 + 3·i)² + b·(-2 + 3·i) + c

0 = 4·a - 12·a·i - 9·a - 2·b + 3·b·i + c

0 = 3·i·(b - 4·a) - 5·a - 2·b + c

∴ b = 4·a

c = 13·a

The general equation becomes

f(x) = a·x² + 4·a·x + 13·a

Also, when x = -1, f(x) = -15

f(-1) = -15 = a·(-1)² + 4·a·(-1) + 13·(-1)

-15 = a - 4·a - 13 = -3·a - 13

-3·a = -15 + 13 = -2

a = 2/3

The general form of the quadratic equation is therefore;

f(x) = 2/3·x² + 8/3·x + 13·2/3 = 2/3·x² + 8/3·x + 26/3

f(x) = 2/3·x² + 8/3·x + 26/3

The minimum value occurs at d(f(x)/dx = 0 = d(2/3·x² + 8/3·x + 13·2/3)/dx = 4/3·x + 8/3 = 0

x = -8/3 × 3/4 = -2

Therefore;

The minimum value of the quadratic function is f(-2) = 2/3·(-2)² + 8/3·(-2) + 13·2/3 = 6

The coordinates of the minimum point, which is the vertex point (h, k) = (-2, 6)

The standard form of the quadratic equation f(x) = a(x - h)² + k, is therefore;

f(x) = 2/3(x - (-2))² + 6 = 2/3·(x + 2)² + 6

f(x) = 2/3·(x + 2)² + 6

Also (h, k) can be gotten from h = -b/2a = -8/3/(2 × 2/3) = -2

k = c - b²/(4·a) = 26/3 - (8/3)²/(4 × 2/3) = 26/3 - 8/3 = 18/3 = 6

k = 6

The standard form of the quadratic equation f(x) = a(x - h)² + k becomes;

f(x) = 2/3·(x + 2)² + 6

answer

a,b, and d

explanation

a reflection angle is more than

but less than;

an angle that measures

is a reflex angle.(true)

an angle that measures less than

is acute.(true)

provided we are considering angles greater or equal to zero.

an acute angle lies in the range.

an obtuse angle is less than 180° but more than 90°

an angle that measures 300° is obtuse (false)

an angle that measures 65° is an acute angle.(true)

no one knows lol

step-by-step explanation: