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\frac{9}{2}x^{2}-27x+45=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(-27\right)±\sqrt{\left(-27\right)^{2}-4\times \frac{9}{2}\times 45}}{2\times \frac{9}{2}}
This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{9}{2} for a, -27 for b, and 45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-27\right)±\sqrt{729-4\times \frac{9}{2}\times 45}}{2\times \frac{9}{2}}
Square -27.
x=\frac{-\left(-27\right)±\sqrt{729-18\times 45}}{2\times \frac{9}{2}}
Multiply -4 times \frac{9}{2}.
x=\frac{-\left(-27\right)±\sqrt{729-810}}{2\times \frac{9}{2}}
Multiply -18 times 45.
x=\frac{-\left(-27\right)±\sqrt{-81}}{2\times \frac{9}{2}}
Add 729 to -810.
x=\frac{-\left(-27\right)±9i}{2\times \frac{9}{2}}
Take the square root of -81.
x=\frac{27±9i}{2\times \frac{9}{2}}
The opposite of -27 is 27.
x=\frac{27±9i}{9}
Multiply 2 times \frac{9}{2}.
x=\frac{27+9i}{9}
Now solve the equation x=\frac{27±9i}{9} when ± is plus. Add 27 to 9i.
x=3+i
Divide 27+9i by 9.
x=\frac{27-9i}{9}
Now solve the equation x=\frac{27±9i}{9} when ± is minus. Subtract 9i from 27.
x=3-i
Divide 27-9i by 9.
x=3+i x=3-i
The equation is now solved.
\frac{9}{2}x^{2}-27x+45=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{9}{2}x^{2}-27x+45-45=-45
Subtract 45 from both sides of the equation.
\frac{9}{2}x^{2}-27x=-45
Subtracting 45 from itself leaves 0.
\frac{\frac{9}{2}x^{2}-27x}{\frac{9}{2}}=-\frac{45}{\frac{9}{2}}
Divide both sides of the equation by \frac{9}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.
x^{2}+\left(-\frac{27}{\frac{9}{2}}\right)x=-\frac{45}{\frac{9}{2}}
Dividing by \frac{9}{2} undoes the multiplication by \frac{9}{2}.
x^{2}-6x=-\frac{45}{\frac{9}{2}}
Divide -27 by \frac{9}{2} by multiplying -27 by the reciprocal of \frac{9}{2}.
x^{2}-6x=-10
Divide -45 by \frac{9}{2} by multiplying -45 by the reciprocal of \frac{9}{2}.
x^{2}-6x+\left(-3\right)^{2}=-10+\left(-3\right)^{2}
Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-6x+9=-10+9
Square -3.
x^{2}-6x+9=-1
Add -10 to 9.
\left(x-3\right)^{2}=-1
Factor x^{2}-6x+9. 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-3\right)^{2}}=\sqrt{-1}
Take the square root of both sides of the equation.
x-3=i x-3=-i
Simplify.
x=3+i x=3-i
Add 3 to both sides of the equation.