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x^{2}-19=3x
Subtract 19 from both sides.
x^{2}-19-3x=0
Subtract 3x from both sides.
x^{2}-3x-19=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(-19\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and -19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-3\right)±\sqrt{9-4\left(-19\right)}}{2}
Square -3.
x=\frac{-\left(-3\right)±\sqrt{9+76}}{2}
Multiply -4 times -19.
x=\frac{-\left(-3\right)±\sqrt{85}}{2}
Add 9 to 76.
x=\frac{3±\sqrt{85}}{2}
The opposite of -3 is 3.
x=\frac{\sqrt{85}+3}{2}
Now solve the equation x=\frac{3±\sqrt{85}}{2} when ± is plus. Add 3 to \sqrt{85}.
x=\frac{3-\sqrt{85}}{2}
Now solve the equation x=\frac{3±\sqrt{85}}{2} when ± is minus. Subtract \sqrt{85} from 3.
x=\frac{\sqrt{85}+3}{2} x=\frac{3-\sqrt{85}}{2}
The equation is now solved.
x^{2}-3x=19
Subtract 3x from both sides.
x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=19+\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}=19+\frac{9}{4}
Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-3x+\frac{9}{4}=\frac{85}{4}
Add 19 to \frac{9}{4}.
\left(x-\frac{3}{2}\right)^{2}=\frac{85}{4}
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{\frac{85}{4}}
Take the square root of both sides of the equation.
x-\frac{3}{2}=\frac{\sqrt{85}}{2} x-\frac{3}{2}=-\frac{\sqrt{85}}{2}
Simplify.
x=\frac{\sqrt{85}+3}{2} x=\frac{3-\sqrt{85}}{2}
Add \frac{3}{2} to both sides of the equation.