Step 1. Write your equation in standard form.
#2x^2 + 16x +42 = 0#
Step 2. Move the constant to the right hand side of the equation.
Subtract #42# from each side .
#2x^2+16x +42 -42 = 0-42#
#2x^2+16x = -42#
Step 3. Divide both sides of the equation by the coefficient of #x^2#.
Divide both sides by 2.
#x^2 +8x = -21#
Step 4. Square the coefficient of x and divide by 4.
#(8)^2/4 = 64/4 = 16#
Step 5. Add the result to each side.
#x^2 +8x + 16 =-21 +16 #
#x^2 +8x + 16= -5#
Step 6. Take the square root of each side.
#x+4 = ±isqrt5#
Case 1
#x_1 + 4 = +isqrt5#
#x_1 = -4+isqrt5#
Case 2
#x_2 + 4 = -isqrt5#
#x_2 = -4 -isqrt5 #
So #x = -4+isqrt5# or #x = -4-isqrt5#
Check: Substitute the values of #x# back into the quadratic.
(a) #x = -4+isqrt5#
#2x^2 + 16x +42 = 2(-4+isqrt5)^2 + 16(-4+isqrt5) +42 = 2(16 -8isqrt5-5) -64 +16isqrt5 +42 = 32 –cancel(16isqrt5) -10 -64 + cancel(16isqrt5) +42= 0#.
(b) #x = 4 - isqrt5#
#2x^2 + 16x +42 = 2(-4-isqrt5)^2 + 16(-4-isqrt5) +42 = 2(16 +8isqrt5-5) -64 -16isqrt5 +42 = 32 –cancel(16isqrt5) -10 -64 - cancel(16isqrt5) +42= 0#.
