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27x^{2}+8x=-3
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.
27x^{2}+8x-\left(-3\right)=-3-\left(-3\right)
Add 3 to both sides of the equation.
27x^{2}+8x-\left(-3\right)=0
Subtracting -3 from itself leaves 0.
27x^{2}+8x+3=0
Subtract -3 from 0.
x=\frac{-8±\sqrt{8^{2}-4\times 27\times 3}}{2\times 27}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 27 for a, 8 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-8±\sqrt{64-4\times 27\times 3}}{2\times 27}
Square 8.
x=\frac{-8±\sqrt{64-108\times 3}}{2\times 27}
Multiply -4 times 27.
x=\frac{-8±\sqrt{64-324}}{2\times 27}
Multiply -108 times 3.
x=\frac{-8±\sqrt{-260}}{2\times 27}
Add 64 to -324.
x=\frac{-8±2\sqrt{65}i}{2\times 27}
Take the square root of -260.
x=\frac{-8±2\sqrt{65}i}{54}
Multiply 2 times 27.
x=\frac{-8+2\sqrt{65}i}{54}
Now solve the equation x=\frac{-8±2\sqrt{65}i}{54} when ± is plus. Add -8 to 2i\sqrt{65}.
x=\frac{-4+\sqrt{65}i}{27}
Divide -8+2i\sqrt{65} by 54.
x=\frac{-2\sqrt{65}i-8}{54}
Now solve the equation x=\frac{-8±2\sqrt{65}i}{54} when ± is minus. Subtract 2i\sqrt{65} from -8.
x=\frac{-\sqrt{65}i-4}{27}
Divide -8-2i\sqrt{65} by 54.
x=\frac{-4+\sqrt{65}i}{27} x=\frac{-\sqrt{65}i-4}{27}
The equation is now solved.
27x^{2}+8x=-3
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.
\frac{27x^{2}+8x}{27}=-\frac{3}{27}
Divide both sides by 27.
x^{2}+\frac{8}{27}x=-\frac{3}{27}
Dividing by 27 undoes the multiplication by 27.
x^{2}+\frac{8}{27}x=-\frac{1}{9}
Reduce the fraction \frac{-3}{27} to lowest terms by extracting and canceling out 3.
x^{2}+\frac{8}{27}x+\left(\frac{4}{27}\right)^{2}=-\frac{1}{9}+\left(\frac{4}{27}\right)^{2}
Divide \frac{8}{27}, the coefficient of the x term, by 2 to get \frac{4}{27}. Then add the square of \frac{4}{27} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{8}{27}x+\frac{16}{729}=-\frac{1}{9}+\frac{16}{729}
Square \frac{4}{27} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{8}{27}x+\frac{16}{729}=-\frac{65}{729}
Add -\frac{1}{9} to \frac{16}{729} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{4}{27}\right)^{2}=-\frac{65}{729}
Factor x^{2}+\frac{8}{27}x+\frac{16}{729}. 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{4}{27}\right)^{2}}=\sqrt{-\frac{65}{729}}
Take the square root of both sides of the equation.
x+\frac{4}{27}=\frac{\sqrt{65}i}{27} x+\frac{4}{27}=-\frac{\sqrt{65}i}{27}
Simplify.
x=\frac{-4+\sqrt{65}i}{27} x=\frac{-\sqrt{65}i-4}{27}
Subtract \frac{4}{27} from both sides of the equation.