Question

solve: 4x^2+3x-2=0 by completing the square

Answer 1

Add 2 to both sides:
4x^2 + 3x = 2

Factor:
4(x^2 + 3/4x + _____) = 2 + _____

To find the blank, divide the coefficient of the x term by 2 to get 3/8; then square it to get 9/64, that's what you put in the first blank. You must add 9/64 to the other side to keep the equation balanced, BUT you must multiply that 9/64 by 4 first to get 9/16 because you're adding 4 times 9/64 to the left side. So...

4(x^2 + 3/4 + 9/64) = 2 + 9/16

4(x + 3/8)^2 = 41/16 (we get 3 / 8 when we divide the 3/4 by 2)
(x + 3/8)^2 = 41/64
\[x+3/8=\pm \sqrt{41/64}=\pm \sqrt{41}/8\]
\[x = -3/8 \pm \sqrt{41}/8 = (-3\pm \sqrt{41})/8\]