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