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x^{2}-3x=y+3
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}-3x-\left(y+3\right)=y+3-\left(y+3\right)
Subtract y+3 from both sides of the equation.
x^{2}-3x-\left(y+3\right)=0
Subtracting y+3 from itself leaves 0.
x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-\left(y+3\right)\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and -\left(y+3\right) for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-3\right)±\sqrt{9-4\left(-\left(y+3\right)\right)}}{2}
Square -3.
x=\frac{-\left(-3\right)±\sqrt{9+4y+12}}{2}
Multiply -4 times -\left(y+3\right).
x=\frac{-\left(-3\right)±\sqrt{4y+21}}{2}
Add 9 to 4y+12.
x=\frac{3±\sqrt{4y+21}}{2}
The opposite of -3 is 3.
x=\frac{\sqrt{4y+21}+3}{2}
Now solve the equation x=\frac{3±\sqrt{4y+21}}{2} when ± is plus. Add 3 to \sqrt{21+4y}.
x=\frac{-\sqrt{4y+21}+3}{2}
Now solve the equation x=\frac{3±\sqrt{4y+21}}{2} when ± is minus. Subtract \sqrt{21+4y} from 3.
x=\frac{\sqrt{4y+21}+3}{2} x=\frac{-\sqrt{4y+21}+3}{2}
The equation is now solved.
x^{2}-3x=y+3
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}-3x+\left(-\frac{3}{2}\right)^{2}=y+3+\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}=y+3+\frac{9}{4}
Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-3x+\frac{9}{4}=y+\frac{21}{4}
Add y+3 to \frac{9}{4}.
\left(x-\frac{3}{2}\right)^{2}=y+\frac{21}{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{y+\frac{21}{4}}
Take the square root of both sides of the equation.
x-\frac{3}{2}=\frac{\sqrt{4y+21}}{2} x-\frac{3}{2}=-\frac{\sqrt{4y+21}}{2}
Simplify.
x=\frac{\sqrt{4y+21}+3}{2} x=\frac{-\sqrt{4y+21}+3}{2}
Add \frac{3}{2} to both sides of the equation.
y+3=x^{2}-3x
Swap sides so that all variable terms are on the left hand side.
y=x^{2}-3x-3
Subtract 3 from both sides.