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-x^{2}+42x+1=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{-42±\sqrt{42^{2}-4\left(-1\right)}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 42 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-42±\sqrt{1764-4\left(-1\right)}}{2\left(-1\right)}
Square 42.
x=\frac{-42±\sqrt{1764+4}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-42±\sqrt{1768}}{2\left(-1\right)}
Add 1764 to 4.
x=\frac{-42±2\sqrt{442}}{2\left(-1\right)}
Take the square root of 1768.
x=\frac{-42±2\sqrt{442}}{-2}
Multiply 2 times -1.
x=\frac{2\sqrt{442}-42}{-2}
Now solve the equation x=\frac{-42±2\sqrt{442}}{-2} when ± is plus. Add -42 to 2\sqrt{442}.
x=21-\sqrt{442}
Divide -42+2\sqrt{442} by -2.
x=\frac{-2\sqrt{442}-42}{-2}
Now solve the equation x=\frac{-42±2\sqrt{442}}{-2} when ± is minus. Subtract 2\sqrt{442} from -42.
x=\sqrt{442}+21
Divide -42-2\sqrt{442} by -2.
x=21-\sqrt{442} x=\sqrt{442}+21
The equation is now solved.
-x^{2}+42x+1=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}+42x+1-1=-1
Subtract 1 from both sides of the equation.
-x^{2}+42x=-1
Subtracting 1 from itself leaves 0.
\frac{-x^{2}+42x}{-1}=-\frac{1}{-1}
Divide both sides by -1.
x^{2}+\frac{42}{-1}x=-\frac{1}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-42x=-\frac{1}{-1}
Divide 42 by -1.
x^{2}-42x=1
Divide -1 by -1.
x^{2}-42x+\left(-21\right)^{2}=1+\left(-21\right)^{2}
Divide -42, the coefficient of the x term, by 2 to get -21. Then add the square of -21 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-42x+441=1+441
Square -21.
x^{2}-42x+441=442
Add 1 to 441.
\left(x-21\right)^{2}=442
Factor x^{2}-42x+441. 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-21\right)^{2}}=\sqrt{442}
Take the square root of both sides of the equation.
x-21=\sqrt{442} x-21=-\sqrt{442}
Simplify.
x=\sqrt{442}+21 x=21-\sqrt{442}
Add 21 to both sides of the equation.