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