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