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-\frac{3}{2}x^{2}-3x-\frac{3}{2}=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{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-\frac{3}{2}\right)\left(-\frac{3}{2}\right)}}{2\left(-\frac{3}{2}\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{3}{2} for a, -3 for b, and -\frac{3}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-3\right)±\sqrt{9-4\left(-\frac{3}{2}\right)\left(-\frac{3}{2}\right)}}{2\left(-\frac{3}{2}\right)}
Square -3.
x=\frac{-\left(-3\right)±\sqrt{9+6\left(-\frac{3}{2}\right)}}{2\left(-\frac{3}{2}\right)}
Multiply -4 times -\frac{3}{2}.
x=\frac{-\left(-3\right)±\sqrt{9-9}}{2\left(-\frac{3}{2}\right)}
Multiply 6 times -\frac{3}{2}.
x=\frac{-\left(-3\right)±\sqrt{0}}{2\left(-\frac{3}{2}\right)}
Add 9 to -9.
x=-\frac{-3}{2\left(-\frac{3}{2}\right)}
Take the square root of 0.
x=\frac{3}{2\left(-\frac{3}{2}\right)}
The opposite of -3 is 3.
x=\frac{3}{-3}
Multiply 2 times -\frac{3}{2}.
x=-1
Divide 3 by -3.
-\frac{3}{2}x^{2}-3x-\frac{3}{2}=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.
-\frac{3}{2}x^{2}-3x-\frac{3}{2}-\left(-\frac{3}{2}\right)=-\left(-\frac{3}{2}\right)
Add \frac{3}{2} to both sides of the equation.
-\frac{3}{2}x^{2}-3x=-\left(-\frac{3}{2}\right)
Subtracting -\frac{3}{2} from itself leaves 0.
-\frac{3}{2}x^{2}-3x=\frac{3}{2}
Subtract -\frac{3}{2} from 0.
\frac{-\frac{3}{2}x^{2}-3x}{-\frac{3}{2}}=\frac{\frac{3}{2}}{-\frac{3}{2}}
Divide both sides of the equation by -\frac{3}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.
x^{2}+\left(-\frac{3}{-\frac{3}{2}}\right)x=\frac{\frac{3}{2}}{-\frac{3}{2}}
Dividing by -\frac{3}{2} undoes the multiplication by -\frac{3}{2}.
x^{2}+2x=\frac{\frac{3}{2}}{-\frac{3}{2}}
Divide -3 by -\frac{3}{2} by multiplying -3 by the reciprocal of -\frac{3}{2}.
x^{2}+2x=-1
Divide \frac{3}{2} by -\frac{3}{2} by multiplying \frac{3}{2} by the reciprocal of -\frac{3}{2}.
x^{2}+2x+1^{2}=-1+1^{2}
Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+2x+1=-1+1
Square 1.
x^{2}+2x+1=0
Add -1 to 1.
\left(x+1\right)^{2}=0
Factor x^{2}+2x+1. 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+1\right)^{2}}=\sqrt{0}
Take the square root of both sides of the equation.
x+1=0 x+1=0
Simplify.
x=-1 x=-1
Subtract 1 from both sides of the equation.
x=-1
The equation is now solved. Solutions are the same.