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Solve for x (complex solution)
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4x^{2}+5x+6=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{-5±\sqrt{5^{2}-4\times 4\times 6}}{2\times 4}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 5 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-5±\sqrt{25-4\times 4\times 6}}{2\times 4}
Square 5.
x=\frac{-5±\sqrt{25-16\times 6}}{2\times 4}
Multiply -4 times 4.
x=\frac{-5±\sqrt{25-96}}{2\times 4}
Multiply -16 times 6.
x=\frac{-5±\sqrt{-71}}{2\times 4}
Add 25 to -96.
x=\frac{-5±\sqrt{71}i}{2\times 4}
Take the square root of -71.
x=\frac{-5±\sqrt{71}i}{8}
Multiply 2 times 4.
x=\frac{-5+\sqrt{71}i}{8}
Now solve the equation x=\frac{-5±\sqrt{71}i}{8} when ± is plus. Add -5 to i\sqrt{71}.
x=\frac{-\sqrt{71}i-5}{8}
Now solve the equation x=\frac{-5±\sqrt{71}i}{8} when ± is minus. Subtract i\sqrt{71} from -5.
x=\frac{-5+\sqrt{71}i}{8} x=\frac{-\sqrt{71}i-5}{8}
The equation is now solved.
4x^{2}+5x+6=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.
4x^{2}+5x+6-6=-6
Subtract 6 from both sides of the equation.
4x^{2}+5x=-6
Subtracting 6 from itself leaves 0.
\frac{4x^{2}+5x}{4}=-\frac{6}{4}
Divide both sides by 4.
x^{2}+\frac{5}{4}x=-\frac{6}{4}
Dividing by 4 undoes the multiplication by 4.
x^{2}+\frac{5}{4}x=-\frac{3}{2}
Reduce the fraction \frac{-6}{4} to lowest terms by extracting and canceling out 2.
x^{2}+\frac{5}{4}x+\left(\frac{5}{8}\right)^{2}=-\frac{3}{2}+\left(\frac{5}{8}\right)^{2}
Divide \frac{5}{4}, the coefficient of the x term, by 2 to get \frac{5}{8}. Then add the square of \frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{5}{4}x+\frac{25}{64}=-\frac{3}{2}+\frac{25}{64}
Square \frac{5}{8} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{5}{4}x+\frac{25}{64}=-\frac{71}{64}
Add -\frac{3}{2} to \frac{25}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{5}{8}\right)^{2}=-\frac{71}{64}
Factor x^{2}+\frac{5}{4}x+\frac{25}{64}. 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{5}{8}\right)^{2}}=\sqrt{-\frac{71}{64}}
Take the square root of both sides of the equation.
x+\frac{5}{8}=\frac{\sqrt{71}i}{8} x+\frac{5}{8}=-\frac{\sqrt{71}i}{8}
Simplify.
x=\frac{-5+\sqrt{71}i}{8} x=\frac{-\sqrt{71}i-5}{8}
Subtract \frac{5}{8} from both sides of the equation.