​ What is the solution set of the equation using the quadratic formula?

Solve for x:

x^2 + 6 x + 10 = 0

　

Subtract 10 from both sides:

x^2 + 6 x = -10

　

Add 9 to both sides:

x^2 + 6 x + 9 = -1

　

Write the left hand side as a square:

(x + 3)^2 = -1

　

Take the square root of both sides:

x + 3 = i or x + 3 = -i

　

Subtract 3 from both sides:

x = -3 + i or x + 3 = -i

　

Subtract 3 from both sides:

x = -3 + i      or      x = -3 - i

Solve for x:

x^2 + 6 x + 10 = 0

　

Using the quadratic formula, solve for x.

x = (-6 ± sqrt(6^2 - 4×10))/2 = (-6 ± sqrt(36 - 40))/2 = (-6 ± sqrt(-4))/2:

x = (-6 + sqrt(-4))/2 or x = (-6 - sqrt(-4))/2

　

Express sqrt(-4) in terms of i.

sqrt(-4) = sqrt(-1) sqrt(4) = i sqrt(4):

x = (-6 + i sqrt(4))/2 or x = (-6 - i sqrt(4))/2

　

Simplify radicals.

sqrt(4) = sqrt(2^2) = 2:

x = (-6 + i×2)/2 or x = (-6 - i×2)/2

　

Factor the greatest common divisor (gcd) of -6, 2 i and 2 from -6 + 2 i.

Factor 2 from -6 + 2 i giving -6 + 2 i:

x = (2 i - 6)/2 or x = (-6 - 2 i)/2

　

Cancel common terms in the numerator and denominator.

(2 i - 6)/2 = -3 + i:

x = i - 3 or x = (-6 - 2 i)/2

　

Factor the greatest common divisor (gcd) of -6, -2 i and 2 from -6 - 2 i.

Factor 2 from -6 - 2 i giving -6 - 2 i:

x = -3 + i or x = (-(2 i) - 6)/2

　

Cancel common terms in the numerator and denominator.

(-(2 i) - 6)/2 = -3 - i:

x = -3 + i        or        x = -i - 3