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x^{2}-6x-3=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(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-3\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-6\right)±\sqrt{36-4\left(-3\right)}}{2}
Square -6.
x=\frac{-\left(-6\right)±\sqrt{36+12}}{2}
Multiply -4 times -3.
x=\frac{-\left(-6\right)±\sqrt{48}}{2}
Add 36 to 12.
x=\frac{-\left(-6\right)±4\sqrt{3}}{2}
Take the square root of 48.
x=\frac{6±4\sqrt{3}}{2}
The opposite of -6 is 6.
x=\frac{4\sqrt{3}+6}{2}
Now solve the equation x=\frac{6±4\sqrt{3}}{2} when ± is plus. Add 6 to 4\sqrt{3}.
x=2\sqrt{3}+3
Divide 6+4\sqrt{3} by 2.
x=\frac{6-4\sqrt{3}}{2}
Now solve the equation x=\frac{6±4\sqrt{3}}{2} when ± is minus. Subtract 4\sqrt{3} from 6.
x=3-2\sqrt{3}
Divide 6-4\sqrt{3} by 2.
x=2\sqrt{3}+3 x=3-2\sqrt{3}
The equation is now solved.
x^{2}-6x-3=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.
x^{2}-6x-3-\left(-3\right)=-\left(-3\right)
Add 3 to both sides of the equation.
x^{2}-6x=-\left(-3\right)
Subtracting -3 from itself leaves 0.
x^{2}-6x=3
Subtract -3 from 0.
x^{2}-6x+\left(-3\right)^{2}=3+\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=3+9
Square -3.
x^{2}-6x+9=12
Add 3 to 9.
\left(x-3\right)^{2}=12
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{12}
Take the square root of both sides of the equation.
x-3=2\sqrt{3} x-3=-2\sqrt{3}
Simplify.
x=2\sqrt{3}+3 x=3-2\sqrt{3}
Add 3 to both sides of the equation.