Find all complex solutions of 4x^2+5x+2=0.
x = [ -5 +/- sqrt(25 -32) ] /8
[ -5 +/- i sqrt 7 ] / 8
-5/8 +/- (i/8)sqrt 7
To find the complex solutions of the equation 4x^2+5x+2=0, we can use the quadratic formula. The quadratic formula states that for any quadratic equation of the form ax^2+bx+c=0, the solutions can be found using the formula:
x = (-b ± √(b^2-4ac))/(2a)
In this case, a=4, b=5, and c=2. Plugging these values into the quadratic formula, we get:
x = (-5 ± √(5^2-4(4)(2)))/(2(4))
= (-5 ± √(25-32))/(8)
= (-5 ± √(-7))/(8)
Since there is a square root of a negative number in the equation, we have complex solutions. To simplify the expression further, we can rewrite the square root of -7 as the square root of 7 times the imaginary unit, i:
x = (-5 ± √(7)i)/(8)
Thus, the complex solutions of the equation 4x^2+5x+2=0 are:
x = (-5 + √(7)i)/(8) and x = (-5 - √(7)i)/(8)