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x^{2}-4x=9
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}-4x-9=9-9
Subtract 9 from both sides of the equation.
x^{2}-4x-9=0
Subtracting 9 from itself leaves 0.
x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-9\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-4\right)±\sqrt{16-4\left(-9\right)}}{2}
Square -4.
x=\frac{-\left(-4\right)±\sqrt{16+36}}{2}
Multiply -4 times -9.
x=\frac{-\left(-4\right)±\sqrt{52}}{2}
Add 16 to 36.
x=\frac{-\left(-4\right)±2\sqrt{13}}{2}
Take the square root of 52.
x=\frac{4±2\sqrt{13}}{2}
The opposite of -4 is 4.
x=\frac{2\sqrt{13}+4}{2}
Now solve the equation x=\frac{4±2\sqrt{13}}{2} when ± is plus. Add 4 to 2\sqrt{13}.
x=\sqrt{13}+2
Divide 4+2\sqrt{13} by 2.
x=\frac{4-2\sqrt{13}}{2}
Now solve the equation x=\frac{4±2\sqrt{13}}{2} when ± is minus. Subtract 2\sqrt{13} from 4.
x=2-\sqrt{13}
Divide 4-2\sqrt{13} by 2.
x=\sqrt{13}+2 x=2-\sqrt{13}
The equation is now solved.
x^{2}-4x=9
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}-4x+\left(-2\right)^{2}=9+\left(-2\right)^{2}
Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-4x+4=9+4
Square -2.
x^{2}-4x+4=13
Add 9 to 4.
\left(x-2\right)^{2}=13
Factor x^{2}-4x+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-2\right)^{2}}=\sqrt{13}
Take the square root of both sides of the equation.
x-2=\sqrt{13} x-2=-\sqrt{13}
Simplify.
x=\sqrt{13}+2 x=2-\sqrt{13}
Add 2 to both sides of the equation.