Skip to main content
Solve for x (complex solution)
Tick mark Image
Graph

Similar Problems from Web Search

Share

3x^{2}+x+12=0
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
x=\frac{-1±\sqrt{1^{2}-4\times 3\times 12}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 1 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-1±\sqrt{1-4\times 3\times 12}}{2\times 3}
Square 1.
x=\frac{-1±\sqrt{1-12\times 12}}{2\times 3}
Multiply -4 times 3.
x=\frac{-1±\sqrt{1-144}}{2\times 3}
Multiply -12 times 12.
x=\frac{-1±\sqrt{-143}}{2\times 3}
Add 1 to -144.
x=\frac{-1±\sqrt{143}i}{2\times 3}
Take the square root of -143.
x=\frac{-1±\sqrt{143}i}{6}
Multiply 2 times 3.
x=\frac{-1+\sqrt{143}i}{6}
Now solve the equation x=\frac{-1±\sqrt{143}i}{6} when ± is plus. Add -1 to i\sqrt{143}.
x=\frac{-\sqrt{143}i-1}{6}
Now solve the equation x=\frac{-1±\sqrt{143}i}{6} when ± is minus. Subtract i\sqrt{143} from -1.
x=\frac{-1+\sqrt{143}i}{6} x=\frac{-\sqrt{143}i-1}{6}
The equation is now solved.
3x^{2}+x+12=0
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
3x^{2}+x+12-12=-12
Subtract 12 from both sides of the equation.
3x^{2}+x=-12
Subtracting 12 from itself leaves 0.
\frac{3x^{2}+x}{3}=-\frac{12}{3}
Divide both sides by 3.
x^{2}+\frac{1}{3}x=-\frac{12}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}+\frac{1}{3}x=-4
Divide -12 by 3.
x^{2}+\frac{1}{3}x+\left(\frac{1}{6}\right)^{2}=-4+\left(\frac{1}{6}\right)^{2}
Divide \frac{1}{3}, the coefficient of the x term, by 2 to get \frac{1}{6}. Then add the square of \frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{1}{3}x+\frac{1}{36}=-4+\frac{1}{36}
Square \frac{1}{6} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{1}{3}x+\frac{1}{36}=-\frac{143}{36}
Add -4 to \frac{1}{36}.
\left(x+\frac{1}{6}\right)^{2}=-\frac{143}{36}
Factor x^{2}+\frac{1}{3}x+\frac{1}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x+\frac{1}{6}\right)^{2}}=\sqrt{-\frac{143}{36}}
Take the square root of both sides of the equation.
x+\frac{1}{6}=\frac{\sqrt{143}i}{6} x+\frac{1}{6}=-\frac{\sqrt{143}i}{6}
Simplify.
x=\frac{-1+\sqrt{143}i}{6} x=\frac{-\sqrt{143}i-1}{6}
Subtract \frac{1}{6} from both sides of the equation.
x ^ 2 +\frac{1}{3}x +4 = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 3
r + s = -\frac{1}{3} rs = 4
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = -\frac{1}{6} - u s = -\frac{1}{6} + u
Two numbers r and s sum up to -\frac{1}{3} exactly when the average of the two numbers is \frac{1}{2}*-\frac{1}{3} = -\frac{1}{6}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(-\frac{1}{6} - u) (-\frac{1}{6} + u) = 4
To solve for unknown quantity u, substitute these in the product equation rs = 4
\frac{1}{36} - u^2 = 4
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 4-\frac{1}{36} = \frac{143}{36}
Simplify the expression by subtracting \frac{1}{36} on both sides
u^2 = -\frac{143}{36} u = \pm\sqrt{-\frac{143}{36}} = \pm \frac{\sqrt{143}}{6}i
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{1}{6} - \frac{\sqrt{143}}{6}i = -0.167 - 1.993i s = -\frac{1}{6} + \frac{\sqrt{143}}{6}i = -0.167 + 1.993i
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.